In 2006 I published this book for the first time, and I’m proud to say that this is the fourth edition of The Relation between Formal Science and Natural Science. In this book, I use Edmund Husserl’s philosophy of logic and mathematics, as well as his semantic doctrine, in order to understand the nature of formal sciences. It posits the existence of ideal meanings and mathematical objects, which are themselves a condition of possibility for any truth and any science whatsoever. It advocates for the search for a criterion to determine the distinction between analytic and synthetic judgments, while rejecting Quine’s arguments against it. At the same time it rejects several antiplatonist options such as Mario Bunge’s fictionalism, and Karl Popper’s semiplatonism, while proposing Husserlian epistemology of mathematics as an alternative, which is essentially a sort of "rationalist epistemology" as Jerrold Katz suggested. Finally, the book criticizes the Quine-Putnam theses, especially the one which states that logic and mathematics can be revised in light of recalcitrant experience. Usually three cases for such revision are constantly presented in this debate: quantum logic, non-euclidean geometry and the general theory of relativity, and chaos theory. I show that none of these a posteriori matters-of-fact have revised any a priori formal fields such as mathematics and logic.
The book’s website has also undergone major surgery, changing it from plain HTML to a Drupal platform. This is how it used to look like:
(Click for Larger Version)
This is how it looks like:
(Click for Larger Version)
You can look at the new website by going to http://uos.pmrb.net. I hope you like it. Any comments or questions about it, please, let me know.
The book is completely available online under different formats. You can download it for free and copy it as many times as you wish just under two conditions: the original work will be preserved verbatim, and no commercial use of it is allowed unless you have reached an agreement with me. Additional to this, because the cover is a derived copylefted version of a GPLed wallpaper in KDE-Look.org, I released the cover and all of its new graphic elements under the GNU GPL as well, and allow people to download it and use it as they wish commercially or non-commercially as long as they comply with that license.
The book is also available for sale for now in Lulu.com.
I hope that this book will help contribute to a clearer understanding about the nature and role of formal sciences such as logic and mathematics, and natural sciences such as physics and biology.
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Up to now, we have seen that psychologism (i.e. conceiving logic as a normative discipline which tells us how to think, or reducing logic to matters-of-fact especially regarding the mind) is unfruitful and implies all sorts of counter-senses, contradictions, and skepticism. Psychologists (i.e. those who promote psychologism) tried their best to provide an objective account for logic and truth, but at the very end, it was all a failure.
Again, Husserl sympathized with that position, because he came from there. He adored his teacher Brentano, but not to the point of sacrificing what he knew had to be true: that logic cannot be reduced to norms of mental operations. As always, in philosophy, many thinkers are misled by their own prejudices.
Psychologism’s Three Prejudices
Perhaps one of the key aspects of Husserl’s criticisms to psychologisms of all sorts has to do with three basic prejudices that permeate them, plague them, and blind these thinkers, and engage them in futile quests. This is the part where Husserl, for all practical purposes, places the nails of psychologism’s coffin.
First Prejudice
According to Husserl, psychologism’s first prejudice can be formulated this way:
The norms and principles which regulate the mind are founded in psychology. Therefore, it is also evident that normative laws of knowledge must be founded on a psychology of knowledge.
This is the core of all psychologism: reducing all knowledge to psychological operations. If logical laws establish the norms of knowledge, then they are norms of psychological operations.
As Husserl already pointed out there are two sides of logic:
- Theoretical aspect of logic: which tells us what is.
- Normative aspect of logic: which tells us what we ought to do.
To understand this distinction, he gives us a good example. Let’s say that someone says something like this:
A good soldier is courageous.
For Husserl, this is a theoretical statement. Don’t be misled by the term "theoretical". It doesn’t mean that it is a conjecture or it is a mere speculation. In Husserl’s sense of the word, this statement tell us what is universally true. In the same sense, the principle of no-contradiction or Modus Barbara, are all theoretical rules of logic.
However, if someone says something like this:
A soldier ought to be courageous.
then, this is a normative statement, because it does not tell us what is, it only tells us what soldiers ought to be.
Now, the question is: which is more fundamental … the theoretical statement or the normative statement? The answer is: the theoretical. A theoretical logical statement tells us what is true no matter what! The normative statement is based on the theoretical statement. The unquestionable self-evident truth that a good soldier is courageous serves as the foundation of the norm that all soldiers ough to be courageous.
Due to how close pure logic and pure mathematics are, Husserl gives us another example to distinguish the theoretical and the normative sides. Let’s take, for example, this formula:
(a + b) (a – b) = a² – b²
When we look at this formula, there is no statement about what we ough to think, just what is. This is itself a theoretical mathematical statement. There is no norm established here (no word "ough" anywhere), nor does it describe any psychological process. However, if we said something like this:
To find the product of the sum and the difference between any two numbers, you ought to establish the difference of their squares.
This is a normative statement, and this norm is based on the theoretical mathematical truth.
By the way, people who wish to Neo-Kantianize Frege have said that for Frege logic is a normative discipline. This is false. Although he doesn’t use the term "theoretical", he fully agrees with Husserl in this very important point. There are many statements which show this very clearly, but one passage from "Thought: a Logical Inquiry" will suffice:
Just as "beautiful" points the way for aesthetics and "good" for ethics, so do words like "true" for logic. … To discover truths is the task of all sciences; it falls to logic to discern the laws of truth. The word "law" is used in two senses. When we speak of moral or civil laws we mean prescriptions, which ought to be obeyed but with which actual occurrences are not always in conformity. … From the laws of truth there follow prescriptions about asserting, thinking, judging, inferring. And we may very well speak of laws of [psychological thinking] in this way too. But there is at once a danger here of confusing different things. People may very well interpret the expression "law of [psychological thinking" with "law of nature" and then have in mind general features of thinking as a mental occurrence. A law of [psychological thinking] in this sense would be a psychological law. And so they might come to believe that logic deals with the mental process of thinking and with the psychological laws in accordance with which this takes place. Error and supertition have causes just as much as correct cognition. Whether what you take for true is false or true, your so taking comes about in accordance with psychological laws. A derivation from these laws, an explanation of a mental process that ends in taking something to be true, can never take the place of proving what is taken to be true. … In order to avoid any misunderstanding and prevent the blurring of the boundary between psychology and logic, I assign to logic the task of discovering the laws of truth, not the laws of taking things to be true or of thinking (Beaney, 1997, 325-326).
What is Husserl and Frege’s point here? Very simple, the norms which we should follow if we wish to find the truth are not founded on psychological laws, but rather in theoretical logical laws. These theoretical laws themselves tell us nothing about mental processes or physical or biological laws which operate the brain or the mind. They only express those formal logical relations to express truth.
Second Prejudice
Psychologism’s second prejudice can be formulated in this way:
Logic is about judgments, reasonings, proofs, probabilities, necessities, possibilities, foundations, consequences and other related cocncepts. But judging, reasoning, finding necessities and probabilities, and the like are all psychological processes. Therefore, logic belongs in psychology.
The problem with this argument is twofold. First, it confuses psychological acts with the actual validity content of these acts. One thing is all the psychological operations that lead me to formulate the sentence: "JFK was killed in 1963". It is quite another different thing to say that the truth expressed in this sentence has a psychological basis. In reality, the truth will not depend at all on psychological acts. It will depend only on two factors:
- Meaning: What the sentence means (its proposition or judgment).
- Referent: If the sentence is fulfilled in a state-of-affairs (fact)
The proposition expressed in "JFK was killed in 1963" is true and will always be true, even if everyone in the future would think it false. In this sense, psychological processes have little to do with the truths which minds are able to grasp. Truths are independent of our minds.
The second problem stems from the fact that logic and mathematics are sister disciplines, logic has mathematics as its necessary ontological correlate, as we have explained before (see this blog post). This means that if logical truths depend inherently on psychological processes, then that means that mathematics does too. Psychologism’s prejudice regarding mathematics is very similar to logic’s: mathematics is about numbers, yet, we need a psychological acts of "counting" to have numbers, or grouping things together, and so on.
Husserl argues that the number five is not the act of counting to five, nor any psychological representation of the number. A number itself is given in a formal structure in a state-of-affairs, but the act of grasping it is altogether different from the number itself. The same goes too for all the laws and principles of arithmetic, geometry and any other mathematical field. These laws are not themselves psychological acts, but truths-in-themselves which we are able to grasp through a psychological act or process. The proposition "2 x 2 = 4" is true and will be forever true regardless of whether tomorrow we were to believe psychologically that "2 x 2 = 5".
From these facts we have to make several distinctions:
- If what we said above is true, then logic and mathematics are ideal sciences, while sciences about matters-of-facts (natural sciences, psychology, anthropology, etc.) are real sciences. The former are a priori, which means that their truths can only be known independently of experience and through reason alone. These would be the realm of relations-of-ideas. On the other hand, real sciences are empirical or a posteriori (based on experience).
- In all knowledge, especially in every sience, we have to distinguish three sorts of interconnexions: first, interconnexions of psychological representations, acts of judging, psychological assumptions, and so on, which occur in the minds of scientists; second, the interconnexions of the objects or objectualities being studied by that science; third, the ideal logical interconnexions among the concepts and truths expressed in scientific theories proposed by such science.
As long as we keep these distinctions in mind, we won’t have any problems and confusions regarding what belongs to logic and mathematics as ideal disciplines, and what belongs to other sciences such as psychology.
Third Prejudice
Psychologism’s third prejudice goes as follows:
If we find a logical or matheatical proposition to be true it is because we find it evident that it is true. Evidence is itself a psychological experience or feeling which is somehow psychologically "attached" to the proposition itself. So the truth or falsity of a proposition necessarily depends on this feeling.
Husserl says that this prejudice confuses once again psychological processes with truth using the notion of "evidence". No logical or mathematical principle or law says anything about the feeling of "evidence" we should all experience. Modus Barbara itself is not equivalent to the "feeling of evidence that Modus Barbara is true", since this logical law says nothing about that.
… There is another confusion besides this, though.
The notion of evidence is not itself a feeling that comes out of the blue. It is more an intellectual assertion that what is being given is true. Evidence is founded on the theoretical side of logic. We find the principle of no-contradiction as evidently true for one reason only: because it is true … period. In mathematics the same. It is impossible for us to psychologically represent in terms of imagery all of the numbers in the decimal form of the number "pi", which is itself an irrational number. Despite this, we know that it is evidently true that there is a trillionth interger of the number pi, yet we are not able to represent it psychologically, nor do we know anything about it (psychologically or otherwise). This is because it is an ideal truth (which escapes all psychological representations) that there is a trillionth interger of the number pi. …. We don’t know what it is, but ideally it is there, it exists. We know this a priori.
Husserl’s Conclusion
If all of these prejudices are wrong, and logic (nor mathematics) can be reduced to psychology, we are forced to establish a distinction between two very different realms:
- Ideal: By "ideal" Husserl means that is existent, but is abstract and independent of the human mind. This is the realm of meanings, essences, true logical relations, numbers, sets, and other categorial forms. This is the realm whose characteristic is that it is atemporal, these truths do not change or are affected by temporal events, in a very real sense they are eternal.
- Real: By "real" Husserl does not mean "existent". Ideal entities, meanings and relations do exist independently of us. What Husserl means by real [or "reell"] is that it is concerned with particular moments in time. This is the realm of physical objects, which persist and change in time, and also of psychological acts which also occur in time.
Basically, psychologism in all of its forms made the mistake of reducing the ideal to the real, something which generates all sorts of contradictions, counter-senses, fallacies, and so on. Only by supposing the independent existence of an abstract ideal realm, we solve philosophically all of these confusions. Platonism is the way to go. 
References
Husserl, E. (2001). Logical investigations. (2 vols.) London & NY: Routledge.
Rosado Haddock, G. E. (2000, October). The Structure of Husserl’s Prolegomena. Manuscrito, 23(2), 61-99.
Hume’s Big Skepticism
Hume was a psychologist, in the sense that he reduced all knowledge to psychological operations. And unlike many of those who came after him, he was very careful in not doubting humanities’ mental faculties. He established a distinction between relations-of-ideas and matters-of-fact. We have discussed that before. He never doubted relations-of-ideas: all circles are round, no matter what. Yet, what about matters-of-fact?
It was here, and not in relations-of-ideas, that Hume’s skepticism was more manifest. A matter-of-fact is, by definition, that sort of truth which is contingent, not logically necessary. It could be otherwise. It is a matter-of-fact that I was born in San Juan, Puerto Rico. But I assure you, if my mom would have taken a trip to Ponce, then it is a possible scenario that I would have been born in Ponce, Puerto Rico … or maybe in New York City … or maybe in Peking … who knows! There are infinite possibilities on how, when, or where I would have been born. There is no logical necessity for me to have been born in San Juan! It is a fact, though.
From the point of view of knowledge, the question is, what are we really given in experience? For Hume, all that we are given from the "outside" world are what he calls impressions: what we see, smell, taste, hear, and touch. For Hume, we are not given actual objects, just impressions. One possible explanation for our impressions is that there are actual objects out very much like the way we sense them. This itself is a matter-of-fact. I remind you that, as a matter-of-fact, this is a just one possibility, and not the only one. There can be vast infinite possibilities of explanations for the impressions we have.
Are these objects "substances" in the Aristotelian sense of the word (an object whose existence is independent of every other object)? Hume was not a practical skeptic … in the sense that if he is standing in the middle of the street, and sees a car approaching him, he will be wise enough to get out of the way. Yet, his problem is one called de jure …. "with what sort of rational right do I derive or infer that there are actually objects (substances) out there when all that we are being given are impressions and nothing more?"
In the same spirit, he also questioned the cause-effect relationship. No one can actually have impressions of something called "cause" (in pure abstraction) and something called "effect" (in pure abstraction). He also goes as far as to debunk Descartes’ notion of the "ego" as the absolute "must exist" for our mental operations to take place. Since the ego is never given in our impressions, and our ego belongs to the realm of matters-of-fact, we cannot actually state its existence.
In other words, as far as it goes with matters-of-fact, we are led to absolute skepticism (theoretical at least). For Hume, there seemed no way out of this. Husserl also learned this from David Hume. And even when Kant tried to circumvent this trying to state that the human mind applies some forms of intuition and pure concepts of understanding, there are several problems with his arguments: first, he is focused solely on human understanding, without taking into consideration any other rational being whatsoever; second, his "pure concepts of understanding" (aka categories) are in reality a mix of Aristotle’s categories along with some conceptualized version of Isaac Newton’s three laws of motion. Husserl took note of that when he called Kantian categories as "mythic" (Logical Investigations, Prol. § 58). Even when Kant wanted to escape skepticism, he could not get away from it fully, hence falling into a sort of relativism, which Husserl would call "specific relativism".
Skeptic Relativism
Psychologism leads to relativism … regardless how much they try to guarantee certain knowledge from their psychologistic prejudices. That is the lesson Husserl taught in the "Prolegomena of Pure Logic" … and still teaches us.
Still … Husserl’s rejection of psychologism as relativism is more refined than Frege’s. Although Frege is right that we should reject psychologists reduction of all logic and mathematics to subjective representations, as we have said before, he had the distinct quality of distorting his opponents’ views, practically reducing all of their doctrines to some gross relativism that many of them never held.
Again, Husserl’s criticism is more effective, because he was much fairer to his opponents. First, he made a distinction between relativists and those who didn’t proclaim themselves as absolute relativists. Second, he made a distinction between the intended purpose of several psychologists, and their unintended logical outcomes. So, even when a particular psychologistic philosophical opinion (or a similar one) would not proclaim itself to be relativistic, Husserl would reveal through logical deduction that their suppositions derive nothing more than relativism, even when the holders of these doctrines don’t intend to.
Husserl recognized that there were two sorts of relativism: individual relativism, and specific relativism.
Individual Relativism
Individual relativism is the form of gross relativism Frege has in mind when accusing all psychologists for being relativists. Yet, as Husserl points out, this form of relativism is so absurd that we should doubt if anyone has taken it seriously. It practically reduces all truth to subjective opinion. In other words, the famous: "What is true for me may not be true for you."
Individual relativism is the opinion that there is no objective truth. As every philosopher knows, this statement is self-defeating. Saying that "There is no objective truth" is equivalent to saying that "It is objectively true that there is no objective truth." There is no truth-in-itself (to use Bolzano’s famous phrase) different or distinct from my mental operations. By establishing all logical principles (such as the principle of no-contradiction) to mere subjective opinions, there cannot be any sort of philosophy built on this. And as Barbara Ehrenreich would say: this sort of relativism, if practiced, cannot even be the basis for any sort of normal conversation.
Specific Relativism
This is a milder form of relativism, which can be called anthropologism. It basically states that any statement is true in relation to a species (hence the word "specific"). In this case, many hold that the set of propositions we call "truth" is in reality truth in relation to humans, because our mental or biological constitution can be different.
For Husserl, Kant tried to avoid individual relativism and the sort of skepticism he so rejected of Hume by adopting an anthropological position: knowledge is "guaranteed" given that humans have such and such faculties.
Let’s see how Husserl rejects and refutes various sorts of anthropologisms:
- Specific relativism says this: each species is capable of judging that it is true what, according to their constitution or their own psychological processes, should be held as true.
Husserl says that there are two ways this anthropological assertion is wrong. First, any statement (proposition) is either true or false if it is fulfilled or not in an actual state-of-affairs. This means that even when other species hold some true statement to be false, then they are holding as true a false belief. In this aspect, both Frege and Husserl are in perfect agreement, one thing is being true, and another thing is holding or believing something as true. Regardless of any species’ constitution, if a species believes that something is true or false, does not mean that it is true or false.
It is a contradiction (or in Husserl’s words "counter-sense") to be talking about a "truth for someone" or a "truth for one species". In his own words: truth is one and identical to any species of rational beings able to grasp it, even if it is for humans, for angels, or for the gods.
- A specific relativist would say: It could be that the words "true" and "false" mean a different thing to another species, since maybe their own mental constitution wouldn’t let them grasp the logical laws which are implied in our own concept of truth: such as the principle of no-contradiction or the principle of the excluded middle.
Husserl responds by saying that if another species uses the words "true" and "false" to mean different things than what we mean, then it is a problem of the meaning of the words "true" and "false". Let us never to confuse name and meaning (as Frege and Husserl pointed out in their respective works — see here for more details here and here). When we are concerned about truth, we are really not concerned about words themselves in their quality as signs, but on what the words mean (i.e. propositions), and their fulfillment in a state-of-affairs (or "facts").
It can be possible that there are extraterrestrial beings who are not able to grasp logical laws as the principle of no-contradiction. If their use of the word "truth" is the same as ours, and still, they wish to negate this particular logical principle, then their negation would be false, even if they thought that such a possibility might be true. However, if their word "truth" means something else altogether, then it is inherently a problem of meaning: they would not be grasping any truth in our meaning of the word. In such case our meaning of the word "truth" is completely unaffected, as are logical laws themselves.
- The specific relativist might say: The constitution of a species is a matter-of-fact, and only matters-of-fact can be derived from other matters-of-fact. The concept of truth and logical laws are matters-of-fact because they are founded on a species’ existence which is itself a matter-of-fact.
Husserl’s response that this is a counter-sense once again. A matter-of-fact is a singular event (a sunset, a star in the sky, a Pres. George Bush who doesn’t know how to spell "nuclear", etc.) In other words, they are temporal events. Yet truths themselves are not subject to the cause and effect relations in time. This confusion happens because psychologism mixes the psychological act of constituting a truth, and the truth-in-itself. Of course we carry out all sorts of mental operations to grasp the truth that "2×2=4". There is absolutely no issue about this. However, the truth "2×2=4" does not depend on us. We can constitute a truth, but we do not create it. Therefore a universal truth is never founded on matters-of-fact, a proposition is only true if it is fulfilled by a matter-of-fact, not that a matter-of-fact "derives" or "infers" a universal truth.
- The specific relativist might argue: If all truth has an exclusive basis on the constitution on the human species, then if there were no human species, then there would not be any truths at all.
This would fall into the same problem as in the case of individual relativism, it is self-defeating because it establishes as objective truth that there are no objective truths at all.
- The specific relativist might argue: It can be possible that given a certain specific constitution, such a constitution would lead a species mentality to conclude as truth that there is no such constitution.
This is another counter-sense. For Husserl, truth is nothing more than a network of true propositions which are necessarily correlated to reality (a whole network of states-of-affairs). What are we to say about this sort of anthropological argument, then? That there is no reality, or that it doesn’t exist except to humans? And, what would happen if all humans disappeared, is reality going to disappear along with it? Definitely, we are moving in contradictions.
It can be possible for a species to have a constitution which can lead it to a false claim. However, it is quite another thing altogether to claim that it would be true to claim that there is no specific constitution because it is itself based on an existent constitution.
By the way, it would not be less absurd if an anthropologist claimed that if such a species recognized the truth of its own specific constitution, then this truth would be founded in such constitution. If they claim that truth is dependent on the species Homo (us!), then this dependence can only be understood causally and according to the laws which rule such causal relation in the constitution. Husserl says that in this case, we would have to claim that the truth "this constitution and these laws exist" would be explained by the fact that they temselves exist, which means that they would be founded on themselves. At the same time, the principles which would agree with such explanation would be identical to these laws themselves. This is non-sense: the constitution would be its own cause, founding itself on laws that would cause themselves by founding themselves on themselves, etc.
- Husserl points out that one further consequence of any argument presented in favor of a relativity of truth (and anthropologism is no exception), is that it implies the relativity of the universe itself. The universe is nothing more than the objectual unity of all states-of-affairs, which are necessarily correlated by all truths about these states-of-affairs. We cannot relativize truth, and at the same time state that there is a universe independent of our own constitution. If there is a truth for a species, then there is a universe for a species. So, if the species disappears, would the universe disappear?
This would be obvious to everyone, but if we reflect a little bit about it, we become aware that our own ego and its psychological acts belong to this universe, which would also mean that every time I say "I exist" or "I have such and such experience", it would be instantly false in a truth-relativistic point of view.
So, question: if our constitution changes, would the universe change along with us to fit our own constitution? And would our constitution, which is part of the universe, change if the universe changes? Nice circle, isn’t it?!
Relativism in General
Husserl, reminding us of Hume, says that all matters-of-fact are contingent: they could be otherwise. If logic is founded on matters-of-fact, then its laws would be contingent, yet they aren’t. They are the foundations for all sciences, and there is a reason for that: any science which rejects these logical laws would be inherently and necessarily self-contradictory. It nullifies itself. We cannot derive any universal logical rule or law from causal and temporal matter-of-fact. Any effort to do so would be self-defeating.
Since psychologism in all of its forms (even in the case of anthropologism) require that logical laws be matters-of-fact, they open themselves to the idea that logical laws are contingent, and there would be absolutely no reason to object any contradictory theory. Remember what Hume taught us: we can be absolute skeptics regarding matters-of-fact, not about relations-of-ideas. The problem with psychologism in Husselr’s time is that it opens the door to being skeptical about absolutely everything, including relations-of-ideas.
References
Hume, D. (1975). Enquiries concerning human understanding and concerning the principles of morals. L. A. Selby-Bigge & P. H. Nidditch (eds.). Oxford: Clarendon Press. (Original work published in 1777).
Husserl, E. (2001). Logical investigations. (2 vols.) London & NY: Routledge.
Rosado Haddock, G. E. (2000, October). The Structure of Husserl’s Prolegomena. Manuscrito, 23 (2), 61-99.
Those who know Husserl very well may ask why did I spend a good deal of time explaining the journey to Platonism describing first his logical and mathematical philosophy, and not with Husserl’s critique of psychologism? If you look at Husserl’s Logical Investigations it is the other way around. Well … that may be so in the structure of Logical Investigations, but in chronological terms, chapter 11 of the "Prolegomena of Pure Logic" was written first, the whole critique of psychologism was written later.
Someone said once (I fail to remember who) that Husserl’s "Prolegomena" represented the most formidable refutation of psychologism ever. I agree. Not even Frege was this good. Frege’s arguments against psychologism and other antiplatonist arguments in general are good, but he has a fatal flaw. If you have been reading these series, especially regarding his review on Husserl, you realize that Frege had this tendency of exaggerating or even distorting his opponents’ opinions. This is unfortunate. Contrary to what people believe, distorting an opponent’s opinion only weakens your own position.
On the other hand, Husserl was extremely fair, and there are many reasons for it. First, he did come from that tradition, so he knows all psychologistic positions, all accross the spectrum. He spends a great deal of time making all sorts of distinctions between psychologists who were more extreme, and those who were not. Second, he usually was very honest about his research and thoroughness. Third, because he was very hard on his errors of the past, errors which went through subtle changes from more extreme to more moderate. And fourth, because he was also critical of philosophers whose positions were close to his own, without distorting their opinions.
What is Logic?
It is difficult to know exactly what logic was in the nineteenth century. So many people held so many positions at that time, that it confused philosophers of every tendency. Psychologism was "in!" at that moment, because, since the time of John Locke, everyone believed that the principles of knowledge could only be achieved by examining our subjective mind. Immanuel Kant tried to overcome the problem by stating that the human mind had faculties and concepts which guarantee knowledge, because we all share these faculties.
Logic, in this sense, was reduced to what people always thought since the beginning: "it is the art of correct thinking." It posits all sorts of rules for us to follow if we want to carry out a thinking process that will lead us to the truth, hence to knowledge. Therefore there are two things which might be said about logic from a psychological point of view:
- Logic is a technique: an instrument which benefits our thinking processes.
- Logic is normative in nature: which means that it establishes the "rules for right thinking"
Husserl will beg to differ on both accounts, and in the "Prolegomena of Pure Logic" he tells us why logic is theoretical: and by this term he does not mean that logic is speculation, it means that logic does not tell us how we ough to think, but tell us formally what is.
When Sciences Go Bezerk
One of the big problems psychologism has is that it wants to submit all forms of knowledge, even formal knowledge, to psychological thinking. If logic is the "art of right or correct thinking" (and notice the word "thinking"), then logic is nothing more than a branch of psychology. In the "Prolegomena", Husserl complains against antipsychologists, because they pretended to beat psychologism while they were conceiving logic as "the art of correct thinking". Hence, when they debated psychologists, psychologism kicked their behind every single time.
Husserl states that antipsychologists are essentially correct, but the reasoning with which they pretend to say that logic does not belong to psychology is seriously flawed by the supposition that logic is a set of rules for us to think.
So, the question is the following. Does logic belong to psychology or not?
In here, Husserl says that we can look at all sciences around us, and see that some are general, and some are more specialized … but nothing too specialized. For instance, we know that there is a science called zoology, yet we don’t see the GRAND field of "science of lions" (or "lionology") or the "science of chairs" ("chairology") anywhere. At least not as a field! Of course, a particular scientist may dedicate his or her whole life to lions and chairs, but it still doesn’t merit a specialized field for all fans of lions and chairs out there!
But what happens when a particular field of science is too broad? What happens if a field occupies issues of another field? What would happen if zoology would incorporate something like botany? Everyone would agree that there is what Husserl would call, in Greek, a "μετάβασις εἰς ἄλλο γένος" (Isn’t Greek pretty? It is pronounced "metábasis eis allo génos") or a "trangression to another genus (field)". Botany is about plants, not animals … therefore it should never be considered a branch of zoology (the science of animals). In this case, botany is a field in its own right.
This is exactly what Husserl thinks about turning logic into a branch of psychology (which is what psychologism is). Psychology is an empirical science, hence, it deals with matters-of-fact. On the other hand, logic is its own field, because it belongs to the realm of relations-of-ideas (or truths-of-reason). So, psychologism would be, for all practical purposes a "μετάβασις εἰς ἄλλο γένος". Psychologism is trying to present as united two fields which are not.
Empirical Consequences of Psychologism
Among psychologists we can count on John Stuart Mill as one of its greatest representatives. Despite the fact that he was considered one of the greatest minds of his times, Frege could not resist the temptation of making fun of him, especially with Mill’s assertion that mathematics is somehow abstracted from sensible experience. Of course, I cannot resist the temptation of sharing with you how Frege made fun of him. This is one of my favorite passages in The Foundations of Arithmetic.
John Stuart Mill … seems to mean to base the science, like Leibniz, on definitions, since he defines the individual numbers in the same way as Leibniz; but this spark of sound sense is no sooner lit than extinguished, thanks to his preconception that all knowledge is empirical. he informs us in fact, that these definitions are not definitions in the logical sense; not only do they fix the meaning of a term, but they also assert along with it an observed matter-of-fact. But what in the world can be the observed fact, or the physical fact (to use another of Mill’s expressions), which is asserted in the definition of the number 777864? Of all the whole wealth of physical facts in his apocalypse, Mill names for us only a solitary one, the one which he holds is asserted in the definition of the number 3. It consists, according to him, in this, that collections of objects exist, which while they impress the senses thus, ⁰0⁰, may be separated into two parts, thus, 00 0. What mercy, then, that not everything in the world is nailed down; for if it were, we should not be able to bring off this separation, and 2 + 1 would not be 3! What a pity that Mill did not also illustrate the physical facts underlying the numbers 0 and 1! (p. 9)
Here is another passage:
[For Mill] it appears that his inductive truth is meant to do the work on Leibniz’s axiom that "If equals are substituted for equals, the equality remains." But in order to be able to call arithmetical truths laws of nature, Mill attributes them a sense which they do not bear. For example, he holds that the identity 1=1 could be false, on the ground that one pound of weight does not alwayss weigh precisely the same as another. But the proposition 1=1 is not intended in the least to state that it does (p. 13).
Although with much less fun, but still remaining highly critical, Husserl sees this same pattern in John Stuart Mill’s work regarding logic. For example, one point of interest of any philosopher of logic is the principle of no-contradiction. This principle states that a proposition and its negation cannot both be true in the same sense at the same time. In symbolic logic we represent it this way:
~ (A & ~A)
Where "A" is any proposition whatsoever ("There is a cat on the roof", "Obama is United States’ president", "The Joker is Batman’s foe"), "~" is the symbol for negation ("no", "not", "it is not the case") and "&" is a conjunction ("and"). In other words, this formula is read like this: "It is not the case that A and not-A". Because Mill is so darn stubborn insisting that all knowledge is abstraction from facts, Husserl criticizes Mill for saying that the principle of no-contradiction is derived from experience.
John Stuart Mill, it is well known, held the principle of [no] contradiction to be ‘one of our earliest and most familiar generalizations from experience’. Its original foundation is taken by Mill to be the fact ‘that belief and disbelief are two different mental states’ which exclude one another. This we know — we follow him verbatim — by the simplest observation of our minds. And if we carry our observation outwards, we find that here too light and darkness, sound and silence, equality and inequality, precedence and subsequence, succession and simultaneity, any positive phenomenon, in short, and its negation, are distinct phenomena, in a relation of extreme contrariety, and that one of them is always absent when the other is present. ‘I consider the axiom in question’, he remarks, ‘to be a generalization from all these facts.’
Where the fundamental principles of his empiricistic prejudices are at stake, all the gods seem to abandon Mill’s otherwise keen intelligence. Only one thing is hard to understand: how such a doctrine could have seemed persuasive. It is obviously false to say that the principle that two contradictory propositions cannot both be true, and in this sense exclude one another, is a generalization from the ‘facts’ cited, that light and darkness, sound and silence, etc., exclude one another, since these are not contradictory propositions at all. It is quite unintelligible how Mill thinks he can connect these supposed facts of experience with the logical law. (Prol. § 25).
Husserl is right: belief (defined as a mental state) is not a proposition, sound and silence are not propositions, light and dark are not propositions, and so on. But here is Husserl’s point: how can John Stuart Mill derive an absolute, necessary, universal logical proposition, from non-absolute, contingent, and singular experiences? What process leads us from one to the other? How can a logical law be a generalized statement from our physical experience in this world? Mill never says how this is so. This is precisely what David Hume criticized about induction.
And this is one of the basic problems with psychologism all accross their spectrum. Even David Hume, an rabid empiricist and skeptic, was far more careful than this!
First Consequence of the Empirical Supposition in Psychologism
Husserl says that psychologists want to legitimize the validity of logical principles as universal and necessary (at least for us), but from a psychological point of view: looking at logical laws as generalizations from sensible experience.
Here is the first reason why it won’t work: From vague foundations you can only derive vague principles ("vague" as opposed to "exact"). The problem with psychology as an empirical science is that its laws can only be probable, never absolutely exact as logical laws are. Since logic is necessarily correlated with mathematics, then also mathematics, which consists of a whole set of exact principles, rules, and laws, would automatically be considered a branch of psychology. So, psychologism is never able to account how it is possible that from the vague laws of psychology we can derive the exact laws of logic and mathematics.
Second Consequence of the Empirical Supposition in Psychologism
Another problem that we have is that psychology is an empirical science, therefore, all of its laws are known by contrasting them with experience. This is not the case in logic, whose rules are known a priori (this means that these rules are known through reason alone, with no reference at all to experience).
For Husserl, the combination of these two consequences generate other unintended consequences. Supposing that all of logic as somehow psychological would mean that no statement can be taken to be absolutely true, but a vague and probable generalization of experience. One of the things Husserl learned from Hume, is that induction cannot guarantee absolute knowledge, only probable ones, because no one can tell you with absolute certainty that similar events in the future will resemble the past. It can always be open to other outcomes. As a result all propositions become probable if their validity relies in operations of the human mind. But think what this would imply: a non-knowledge! Exactly the opposite of what psychologists are searching. Take this proposition, which, by definition, would be only probable (never absolute):
All knowledge is only probable.
Let’s establish this a (let’s say) 90% of probability. Fair enough! Now, through a process of iteration, I can say:
The statement ‘All knowledge is probable’ is probable.
The statement ‘The statement "All knowledge is probable" is probable’ is also probable.
The statement ‘The statement "The statement "’All knowledge is probable"’ is probable" is also probable’ is also probable.
.
.
.
And we could continue ad infinitum, endlessly, each with its own probability. When that happens, the probability of the original proposition being true converges to 0%. In other words, unintentionally, psychologism by its own theory, denies any knowledge whatsoever. (I know that Husserl must have had fun when he wrote this critique).
And even if psychologists wanted to make logical laws as natural laws, we have to ask, how is this statement justified at all at any level of psychologistic literature? The reason for this confusion is that many psychologists actually confuse the causal laws of nature with the non-causal logical laws, even though they try the best to derive one from the other.
Even if they want to define logic as the art of correct thinking, and define "correct" thinking as the way people "normally" think, their feet are too deep in the mud. How many people don’t have so many misconceptions of reality that they actually believe in contradictory things? And how do you place a probable value to that? What guarantees you that the exception, not the rule, are the ones thinking straight?
Third Consequence of the Empirical Supposition in Psychologism
The third consequence of psychologism is that it would interpret logic in terms that are really strange and foreign to it. If logic told us normative principles of thinking, they would have at least some psychological content: some reference to thought processes. Yet, we can find absolutely no trace of matter-of-fact, sensible experience, or thought processes anywhere in logical laws.
Normative statements say: "… you ough to …" Now let’s take Modus Barbara, a well known logical rule.
If all As are Bs
If all Bs are Cs
———————–
Then: All As are Cs
So far so good! Well … may I ask you, my dear reader, where is the "ough" part of this logical rule? The word "ough" is nowhere to be found! For Husserl, this is a theoretical rule … it tells us what is, not how we ough to think. Now if I said something like: "If it is a fact for you that all As are Bs, and that all Bs are Cs, then you ough to think that all As are Cs." … then this statement is normative and does tell us how to think. This statement does have psychological content.
If we want to make logical laws be empirical in some way, then we should look at what empirical or natural laws really are. Empirical laws (conceived in a nomological-deductive manner) or natural laws, along with certain circumstances, do explain phenomena. Therefore, they all have empirical content. Even the most abstract physical laws, which seem to resemble in so many aspects logical and mathematical laws, cannot justify themselves without some reference to experience.
For Husserl, from the point of view of knowledge, it is clear that the laws of natural science which refer to facts, are are fictions with fundamento in re (founded on the thing), in other words, founded on the objects of experience. They are, in Popperian terms, conjectures which have to be tested in experience. … Interesting! These physical theories are just a very small set of an infinite horizon of possible theories which may fit experience. We choose the ones we have because they are the simplest ones which can explain all the phenomena we witness.
Yet, none of this content can be found in any logical law, nor in mathematics. There is no psychological (nor any other empirical) matter-of-fact in a statement like "3 > 2": it doesn’t talk about psychological processes, nor oranges, nor computers. Psychologism is off the mark in this one.
For Husserl, it is undeniable that our knowledge of logic and mathematics are the results of mental processes … but be careful! The fact that there are psychological processes to know "3 > 2" does not mean, that logical and mathematical statements in some way refer to psychological matters-of-facts. Psychologists make this confusion constantly. One thing is the psychological activity of counting to "3", and another the 3 itself.
Combine all of these three consequences of psychologism, and what do you have in the end? Something very simple. If psychologism is true, then no knowledge is possible… and if this is true, then we are led to skepticism. That will be the subject of our next blog post.
References
Frege, G. (1999). The Foundations of Arithmetic. Evanston: Northwestern University Press.
Husserl, E. (2001). Logical investigations. (Vols. 1-2). (J. N. Findlay , Trans.) NY: Humanities Press. (Original work published 1900/1901, 2nd ed. 1913).
The "Duhem-Quine Thesis" … A Misnomer in Philosophy if I Ever Heard One
Some people have asked me, after knowing that I am a platonist, what am I to make of the "Duhem-Quine Thesis". When that happens, I point out to them that the "Duhem-Quine Thesis" has a lot in common with unicorns in a very important aspect: there is no such thing!
Pierre Duhem said one thing, and W. V. O. Quine said a very different thing. Let’s start with Quine. For Quine the whole of knowledge is precisely that, a whole unit, a whole network of propositions which interdepend on one another, always subject to revision in light of recalcitrant experience. Yet, despite the fact that many people rushed to embrace this proposal in order to reject the analytic and synthetic distinction, other people have some problems with this. To summarize Quine’s proposal: nothing is sacred, everything is subject to revision in light of recalcitrant experience.
Pierre Duhem (1861-1916)
Duhem was far more careful than that, as philosopher of science Donald Gillies has pointed out. Pierre Duhem did recognize that in physics … and only in physics … there seemed to be some sort of network of propositions which interpret particular phenomena. Let’s say, for instance, that I wish to throw a rock at a certain angle upwards so that it lands some 10 feet away from me. I can predict which amount of force will be necessary and the energy required for that rock to land 10 feet from me. …
And that’s the trick … isn’t it? To make that "sole" hypothesis, I have to suppose a whole baggage of Newtonian theory: theory of mass, of force, of energy, of how are these concepts related to something like velocity and speed, or acceleration, etc. Then I’ll have to include concepts like gravity, gravitational constant, the relationship between gravitational acceleration and masses, etc.
So if you carry out an experiment, you are not just testing one little teeny weeny hypothesis. Essentially you are testing a whole theoretical group of scientific suppositions and statements which interpret these phenomena, and tell you how to run your experiment. As Duhem said: "An Experiment in Physics Is Not Simply the Observation of a Phenomenon; It is, Besides, the Theoretical Interpretation of This Phenomenon" (Duhem 1905/1991, p. 144).
Don’t believe me? Here, let Duhem explain it to you:
Go into this laboratory; draw near this table crowded with so much apparatus: an electric bettery, copper wire wrapped in silk vessels filled with mercury, coils, a small iron bar carrying in mirror. An observer plunges the metallic stem of a rod, mounted with rubber, into small holes; the iron oscillates and, by means of the mirror tied to it, sends a beam of light over to a celluloid ruler and the observer follows the movement of the light beam on it. There, no doubt, you have an experiment; by means of the vibration of this spot of light, this physicist minutely observes the oscillations of the piece of iron. Ask him now what he is doing. is he going to answer: "I am studying the oscillations of the piece of iron carrying this mirror?" No, he will tell you that he is measuring the electrical resistance of the coil. If you are astonished and ask him what meaning these words have, and what relation they have at the same time perceived, he will reply that your question would require some very long explanations, and he will recommend that you take a course in electricity. (Duhem 1905/1991, p. 145).
So, again, the problem anyone ignorant in physics has is that he or she will never understand what is going on in an experiment, hence will not have the necessary background to interpret it. As Duhem argues very well, if I’m ignorant of the life of the sea, I could not understand: "All hands, tackle the halyard and bowlines everywhere!" Regardless of my own particular understanding of this order, the men on the ship understand it very well and carry out those orders (Duhem 1905/1991, p. 148).
Experiments are only possible, if there is a previous scientific theory to interpret such results (Duhem 1905/1991, pp. 153-158).
But notice that, for Duhem, unlike Quine, he restricts it to physics. This is not applicable to physiology or other fields … and much less to mathematics and logic. I agree with Duhem to a certain extent, but some aspects of physiology have much theoretical baggage behind it too, other aspects of it don’t.
So, when people talk about the "Duhem-Quine Thesis" is in reality Quine’s thesis, not Duhem’s.
Husserl’s Conception of Science
Edmund Husserl was not a philosopher of science, but his philosophy was definitely inspired by physics, a discipline he so admired. In part, his philosophical enteprise, even his phenomenological research, was directed to legitimize science, for its incredible value to society.
Yet, he knew that for science to be reliable it had to obey logical and mathematical laws. How did Husserl think science builds its theories and interprets observations? Here is a direct quote from his Logical Investigations.
"Empirical laws" have, eo ipso, a factual content. Not being true laws, they merely say, roughly speaking, that certain coexistences or successions obtain generally in certain circumstances, or may be expected, with varying probability, in varying circumstances. (Vol. I. Prol. § 23).
What is he saying here? First of all, science operates according to "empirical laws". These laws are not "true laws" in the sense that they are not as universal and necessary as logical and mathematical laws are. They seem to be laws that are at least valid in our own universe, in our own reality. However, following Leibniz, unlike natural laws, Husserl regarded logical and mathematical laws to be valid in every possible world.
These empirical laws, posited by scientific theories, seem to be interconnected logically among themselves. If we could represent it in some way, let’s do it this way:
L1 & L2 & L3 & L4 . . . Ln
According to Husserl, these laws do not operate by themselves, because they have "no factual content". They establish what laws operate in the universe, but they don’t tell us about actual events. Why is that? Because events operate according to these laws and "varying probability, in varying circumstances"! In other words, we could represent Husserl’s views on how science explains phenomena this way: science formulates theories which posit some regularities called "laws" (L), and that these laws along with certain circumstances (C), will lead to an explanation of phenomena (P).
L1 & L2 & L3 & L4 . . . Ln
C1 & C2 & C3 & C4 . . . Cn
—————————————-
P
Now … if you are versed in philosophy of science, you will be very surprised to see Husserl formulating in a sketchy way Carl G. Hempel’s deductive-nomological scheme. as he proposed it in the 1940s. In a very short passage, Husserl shows how way ahead of his time he was!
The Formal Components of Scientific Theories
Science, like every other field, is a large theoretical group of propositions. As we have seen in Husserl’s theory of sense and referent, scientific propositions, like all propositions, refer to states-of-affairs. Let’s examine propositions for a moment.
Acts of Meaning
The only question we never really answered in these series is how do we formulate propositions? Which acts of consciousness intervene in these process? Remember, our primal constitution is of states-of-affairs. For example, we can constitute a white sheet of paper on the desk. However, by other acts of consciousness, I mentally constitute another different sort of form to refer to that state-of-affairs. That mental act is what Husserl calls a meaning act, which makes possible for me to say "There is a white sheet of paper on the desk". Just as formal-ontological cateogories, the word "is" does not have any sensible correlate (such as the sheet of paper itself), but it establishes the existence of such a white sheet in a particular manner. In the same way, I can constitute Megan taller than Mary, but by a meaning act I can propose that "Megan is taller than Mary".
And the word "is" in this context does correlate with a categorial form, but not it is not a formal-objectual category … but a meaning category. What meaning categories do is to structure objectualities in such a way that it is possible to communicate what we want to propose in a meaningful manner. Let me give you an example of what I mean.
Imagine someone who would tell you "table Zingale the outside sits porch at". Of course, this is not a meaningful thing to say … in fact it is not a statement at all, since statements are meaningful. Yet, if you follow the rules of grammar, then you can say that "Zingale sits at the table outside the porch" … interesting place to sit.
Husserl says exactly the same things. Meaning categories let us arrange objectualities and actions in a meaningful proposition. For these propositions to be meaningful, this arrangement has to follow universal and necessary grammar rules for meaning. We are not talking here about the rules of grammar in a specific language. Even when in English the verbs are in the middle, and in German at the end, it makes no difference for Husserl. The grammar he is talking about has to do with the way meaningful propositions are arranged in the abstract sense. This is a realm which linguistics knows very well … as is the idea of a Universal Grammar proposed by Noam Chomsky. His point is the same as Husserl’s … underlying every language on Earth there are some basic structures shared to express states-of-affairs. The only difference between Chomsky and Husserl is that the former established it in naturalistic terms, the latter in a priori terms.
Like all propositions, scientific propositions have material meanings (concepts or meanings of proper names which refer to objects) and formal concepts (which refer to categorial forms). Through categorial abstraction, let’s get rid of all of the material concepts and states-of-affairs, and what do you have? The form of the proposition in its purity, or meaning categories, which include, but are not limited to:
- Subject – Predicate Structure
- Forms of Plural
- Conjunction ("and")
- Disjunction ("or")
- Implication ("if … then ….)
- Negation ("no", "not")
Also, if these propositions are associated in a deductive or logical manner (like scientific laws are), we are able to see also these deductive relations among them in their purity, without appealing to any sort of sensible content.
The Relationship between Formal Logic and Mathematics: a Mathesis Universalis
Again, science is made up of propositions, which refer to states-of-affairs. Let’s remember that if a proposition is true, it is because it is fulfilled in a state-of-affairs, or it has a state-of-affairs as its correlate. Yet, if science proposes a set of logically and deductive related propositions, they correlate with a whole network of states-of-affairs.
Formalize propositions and states-of-affairs through categorial abstraction, and you will have meaning categories in their purity deductively and logically interconnected with one another on the side of propositions, and correlated with these are formal-objectual categories in their purity on the side of states-of-affairs.
These meaning categories are the basis of formal logic, while formal-objectual categories are the basis for mathematics. We can see here the relationship between the two … but how do they integrate in a "mathesis universalis"? Here is how Husserl solved the problem. He divided the correlation of formal or pure logic on the one hand and pure mathematics on the other in three different strata.
First Logical and Mathematical Stratum
Like we have seen, logic is made up of meaning categories, forms of plural, conjunction, disjnction, implication, negation, subject-predicate structures, and so on. This is the stratum of a priori universal grammar, where meaning categories arrange objectualities into meaningful propositions. This is called by Husserl a morphology of meanings (in other words, how meaning categories "shape" propositions). This stratum is ruled by a priori laws which he called laws to prevent non-sense.
On the side of mathematics we have formal-objectual categories, which formally "shape" and structure objects in states-of-affairs. Husserl called this a morphology of intuitions or morphology of formal-objectual categories. Here we find formal-objectual categories such as: cardinal numbers, ordinal numbers, sets, relations, parts-whole, and so on.
Second Logical and Mathematical Stratum
On top of this first stratum, we find that propositions can be organized deductively according to simple syllogisms (as Aristotle proposed). For example, take Modus Barbara:
If all animals are mortals
If men are animals
—————————————–
Then all men are mortals
Let’s get rid of the material components, and we will have this simple form of deduction:
If all As are Bs
If all Bs are Cs
————————-
Then all As are Cs
In this stratum, truth is not really a concern, only the forms of deduction count, just like the one expressed in this case.
These deductive laws are a priori, and they are called by Husserl laws to prevent counter-sense (contradictions).
In a still upper level (not yet the third), we integrate in this logical stratum the notion of truth and similar concepts, where only true propositions are concerned. He called it the logic of truth.
On the side of mathematics, we find a whole set of disciplines founded on the formal-ontological categories on the first level. For example, with the notion of cardinal number, and other sorts of numbers, we can develop arithmetic as a discipline. On the basis of sets, we can develop set theory. On the basis of part-whole categories, we can develop mereology … and so on. For these disciplines to progress, they use deductive laws of logic in this logical stratum. So, we start to see a gradual integration of mathematics and logic.
Third Logical and Mathematical Stratum
Then there is a third logical level where pure logic becomes a theory of all of the forms of theories or a theory of deductive systems. In this level, a logician is not limited to the simple logical deductions we find in the second stratum, but he or is free to posit and explore exhaustively other formal deductive systems. The only rule of this game is to preserve truth in virtue of their deductive forms.
In fact, that is what logicians today actually do. Husserl had no idea at the time how this was, since, like Frege, he blamed psychologism for this lack of development at that time. Actually, ever since Frege, there was an explosion of search for alternative deductive systems. In this sense, not being a logician himself, Husserl did foresee what logic would become as time went by.
On the side of mathematics, mathematics becomes a theory of manifolds, a mathesis universalis, where a mathematician can posit other mathematical concepts (as in Husserl’s time: negative roots, sets, fractions, and so on) or even add and subtract some mathematical axioms (such as the elimination of the axiom of the parallels, or the creation of rules regarding negative roots or negative numbers, how to handle fractions, etc.). These mathematicians would explore exhaustively all of the consequences of these systems, whose validity will depend on absolute consistency. The logical deductive systems developed at this logical level can be used in this stratum too. For Husserl, the completeness of mathematics should be kept in mind in this stratum. Today, Gödel’s theorems ruined any expectation on the completeness of mathematics, but in a way it can be kept as a sort of Kantian ideal guide for this theory of manifolds to operate fully.
In this way, each logical stratum has as its ontological correlate a mathematical stratum. The correlation is not perfect, but they do explain the relationship between logic and mathematics. Here below is a graphical representation of everything we have just explained.
Some Interesting Facts …
Rudolf Carnap is known to have made the distinction between formation rules and transformation rules, and this went through history of logic as being his particular contribution to the subject. But we know as a matter of fact that Carnap was pretty much familiar with Husserl’s Logical Investigations, and used Husserlian terminology extensively in both of his first major philosophical works: On Space and The Logical Structure of the World. However, due to his relationship with members of the Vienna Circle who were pretty much anti-Husserl, he wanted to water down Husserl’s contributions to his philosophy, especially in The Logical Structure of the World.
Carnap made this distinction between formation rules and transformation rules in his Logical Syntax of Language, yet it smells that it is one of those occasions he never attributed Husserl the original distinction in Logical Investigations. The laws to prevent non-sense are the Carnapian formation rules, while the laws to prevent counter-sense are the Carnapian transformation rules.
Wadda ya know!
Unfortunately, many scholars who have focused too much on Husserl’s phenomenological doctrine have ignored completely this aspect of Husserl’s work I’ve just presented above. This aspect of Husserl’s philosophy has been worked out by Verena Mayer, Dallas Willard, Claire Ortiz Hill, Guillermo E. Rosado Haddock, Rudolf Bernet, Iso Kern, Eduard Marbach, among other scholars (though very few). For all those interested, the first version of this doctrine appears in Logical Investigations, in chapter 11 of the "Prolegomena to Pure Logic", on his Fourth Investigation, and the Sixth Investigation. For the most detail version of it (the most elaborated one we have to date), we can find it in Husserl’s Formal and Transcendental Logic.
References
Bernet, R., Kern, I., & Marbach, E. (1999). An introduction to Husserlian phenomenology. IL: Northwestern University Press.
Duhem, P. (1991). The aim and structure of physical theory. US: Princeton University Press. (Original work published in 1905).
Gillies, D. (1993). Philosophy of science in the twentieth century: four central themes. Oxford & Cambridge: Blackwell.
Hill, C. O., & Rosado, G. E. (2000). Husserl or Frege? Meaning, objectivity and mathematics. IL: Open Court.
Husserl, E. (1969). Formal and transcendental logic. (D. Carns, Trans.) The Hague: M. Nijhoff. (Original work published in 1929).
Husserl, E. (1973). Experience and judgment. (J. S. Churchill & K. Ameriks, Trans.). London: Routledge & Kegan Paul. (Original work published in 1939).
Husserl, E. (1998). Ideas pertaining to a pure phenomenology and to a phenomenological philosophy. The Hague: Kluwer Academic Publishers. (Originally published in 1913).
Husserl, E. (2001). Logical investigations. (Vols. 1-2). (J. N. Findlay , Trans.) NY: Humanities Press. (Original work published 1900/1901, 2nd ed. 1913).
Rosado Haddock, G. E. (2000, October). The Structure of Husserl’s Prolegomena. Manuscrito, 23 (2), 61-99.
Rosado Haddock, G. E. (2006). Husserl’s philosophy of mathematics: its origin and relevance. Husserl Studies, 22, 193-222.
Rosado-Haddock, G. E. (2008). The young Carnap’s unknown master: Husserl’s influence on Der Raum and Der logische Aufbau der Welt. US: Ashgate.
Mayer, V. (1991). Die Konstruktion der Erfahrungs Welt: Carnap und Husserl. In W. Spohn (Ed.) Erkenntnis Orientated. (pp. 287-303). Dordrecht: Kluwer.
Mayer, V. (1992). Carnap un Husserl. In D. Bell & W. Vossnkuhl, (Eds.). Wissenschaft und Subjectivität. Berlin: Akademie Verlag.
Problems with Sets
When set theory was first formulated by Georg Cantor and elaborated later by Ernst Zermelo, they noticed that the way the whole system was formulated gave way to paradoxes. Usually a paradox comes up when the system deductively lets certain cases derive contradictory propositions. One famous paradox (although not a set theory paradox) is called "The Liar Paradox": when a liar says "I’m always lying", is he lying or not? If he is lying then that means that he tells the truth (not always lying), but if he is telling the truth, then he is lying. It is a paradox so common since antiquity, that this appears in the Bible (Titus 1:12).
Set theory had two major paradoxes at the time.
1. Cantor’s Paradox
Georg Cantor and other mathematicians developed the concept of power set. It is the set of all the subsets of a given set. …. Ok! Ok! … I know, it sounds confusing, but let me give you an example. Let’s say that we have this set, let’s call it S:
S = {x, y, z}
Then the power-set (P) of S is the following:
P(S) = { {}, {x}, {y}, {z}, {x,y}, {x,z}, {y,z}, {x,y,z} }
In other words, we can take set S, and the power-set will tell you all of its possible subsets (lower order sets). The first one will always be the "empty set" (that is, a set with no elements), then each element of the set can serve also a subset. Then every possible pair of elements is a subset. Finally, all three elements themselves can be also a subset. Simple, right?! The power set is usually bigger than the original for this reason.
Now, here is the problem with what has come to be known as naïve set theory, with the Cantor Paradox:
Naïve set theory states that it is possible to form the set of all sets. If it is the "set of all sets", then there should not be any bigger set than that … right? But what about the power set? The power set of that set of all sets is by definition greater than the set of all sets … which would mean that the set of "all" sets is not really the set of all sets anymore
. Hence a paradox occurs.
2. The Zermelo-Russell Paradox
Frege published the first volume of its Basic Laws of Arithmetic in 1893 as the beginning of his proof that arithmetic is derivable from logic. Just when he was about to publish the second volume in 1903, he received a letter from Bertrand Russell telling him that his work was wonderful, but it had a problem: it allowed for a paradox, which is today known as the Russell Paradox. Today we know that Ernst Zermelo discovered it first and independently from Russell, hence I call it "the Zermelo-Russell Paradox". This paradox blew up Frege’s logicist enterprise to oblivion. I wouldn’t have liked to be in Frege’s shoes. Imagine that you carry out an entire life trying to prove something, and then find out that the enterprise itself was in vain in the end, despite Frege’s own contributions to logic, mathematics and semantics.
But what is this paradox about?
Let’s point out the fact that there are sets which are not part of themselves, or that they are not elements of themselves. What do I mean by that?
Let’s imagine a set of all cats. This necessarily goes from the black cat that you are so afraid to find on the street, to the Cheshire cat who tormented Alice. Imagine all cats grouped together in a set. That set itself is not a cat … right? Therefore, the set of all cats it is not part of itself.
Now, let’s imagine a set of all tables; this set does not form part of itself either … because the set of tables is not itself a table.
And we could go on, the set of all chairs, all TVs, all Presidents of the U. S. … you name it!
Now take all of these sets, and form the set of all sets which do not form part of themselves. This means, take the set of all cats, the set of all tables, the set of all TVs, etc. .. and form this huge "mega-set" of all the sets which do not form part of themselves.
This might seem plausible, right? Now, here is the paradox: Does this "mega-set" form part of itself or not? If it does not form part of itself, then by definition it should form part of itself. But if it does form part of itself, then, by definition, it does not!
After reading about these two paradoxes, I imagine you saying something like: "Wow! Apparently mathematicians have nothing else to do with their time!" Yet, these are not minor problems. … Ask Frege! He’ll tell you all about it.
Epistemology of Mathematics
As I said in my previous blogs, Husserl was not only concerned about logic or mathematics, but he was also concerned about knowledge.
His answer to the problem of knowing mathematical objects has a lot to do with the reasons why he left psychologism … and submitted all sorts of criticisms to empiricism and naturalism in general (which we’ll see in future blog posts). For now, suffice to say, that part of the reason why he left behind all sorts of naturalistic accounts for science is that the theory of knowledge was inadequate. In Ideas pertaining to Pure Phenomenology and to a Phenomenological Philosophy (1913) he criticizes psychologists (proponents of psychologism), naturalists and empiricists.
First of all, Husserl recognizes the enlightened spirit of philosophical naturalists in general, especially when they wish to eliminate all sorts of mysticisms and superstitions from philosophy. He says: "Hey, I get it! And guess what? … I totally agree". But in so doing, they do too much trying to extirpate all sorts of necessary aspects to all knowledge: such as, for instance, essences.
Naturalists are essentially anti-essences. They wish to extirpate essences because they form part of a Platonic heritage, which appeared also as part of Aristotle’s metaphysics. Aristotle apparently sunk the Middle Ages into … the Middle Ages , or in pure darkness of ignorance … only to be rescued by rational and, especially, empirical and naturalistic thinking. Great!
Yet, Husserl points out that if you take a very good look to scientific theories and the way they have succeeded, they all rest in essences. What is an essence anyway? An essence is what is conceptually or logically necessary and always true or always false, no matter what. Logical and mathematical schemes are a network of essential relations among propositions made in scientific theories, which refer to a network of objects (in this case, observable phenomena). In the case of usual geometry, where we use points, lines, space, planes, and so on, we establish also material (not only formal) relationships among these concepts, and science also depends on this.
As the Emperor of Star Wars, we could say about essences: "There is no escape!"
Also, as we have stated in our earlier post, observable objects are not the only things given to us, but also their formal relations, the famous formal-objectual categories.
Therefore, the problem with naturalist epistemology is that it is incomplete, and it has to account for our knowledge of essences and formal categories.
Mixed-Categorial Acts
Husserl bases his theory of knowledge on intentional acts. Remember that intentionality is an act of our own consciousness which directs itself to an object.
Yet, as we have stated before, what is given to our consciousness is always a state-of-affairs. They are objectualities; they are referents of our intentional acts. Yet, as we have seen in our previous blog post, states-of-affairs have two very important components: material content (sensible content), and formal arrangement (formal-objectual categories). How are these constituted?
Husserl’s theory of knowledge is based on what he calls intuitions. An intuition is that aspect of our consciousness which gives us whatever our intentional acts are directed to. Let me explain this in English. For example, if I’m looking at the screen, I have an intuition of the screen being in front of me: it is given to me at once, it is there, and my thinking is directed to it (that is my intentional act).
For Husserl there are two sorts of intuitions involved in the constitution of an objectuality or state-of-affairs:
- Sensible Intuition: This is what makes possible the constitution of objects we see, hear, taste, smell and touch. He further identifies two sorts of sensible intuition: sensible perception, which involves the intuition of objects being given to us "in person" (so-to-speak), like this computer screen in front of me; and sensible imagination (my constitution of objects in my imagination: fairies, imaginary tables, crystal castles, and so on).
- Categorial Intuition: This is what makes possible the constitution of formal-objectual categories: numbers, sets, part-whole, relations, and so on. For him, there is also a distinction between categorial perception (when categorial forms are founded on objects "in person") and categorial imagination (when they are founded on imaginary sensible objects).
Each objectuality is the result of what Husserl calls an objectual act, the mental act of turning a state-of-affairs into our object of our consciousness. Each objectual act our consciousness carries out in the world or in our imagination is what he called: mixed-categorial acts. What does that mean? It means that these acts constitute a state-of-affairs in which we are given sensible objects along with their formal-objectual categories. I perceive three pencils, or a set of books, or Megan being taller than Mary, and so on. All of these constitutions of sensible plus formal states-of-affairs are the result of mixed-categorial acts.
Categorial Abstraction
Yet, when we deal with mathematics … pure mathematics … we do not deal with three pencils or a set of books. We deal with the number three itself, or the set. We don’t have to deal with: "two books and two books are four books in total" … but with "2 + 2 = 4". In fact, sometimes we deal with equations such as "x + 1 = 3" or even "x + y = y + x" or {x, y, z} and so on.
Here, Husserl introduces the concept of categorial abstraction, where intentionally we disregard all sensible objects given in sensible intuition, and just deal with the categorial forms themselves. This is what Husserl calls a pure-categorial act, with which we constitute pure categorial forms, they become our object of our consciousness. This is also called formalization.
So, when we talk of mathematical intuition in Husserl’s case, we are talking about categorial intuition plus categorial abstraction. That’s how we know abstract objects such as numbers, sets, and so on.
Eidetic or Essential Intuition
Yet there is a third form of intuition which Husserl calls eidetic or essential intuition: an intuition of essences. What does this intuition do? It gives us essential relations at once, we are able to recognize through understanding the universality and necessity of what is being given. Let me give you a very obvious instance of how this happens:
"x + y = y + x"
This equation is constituted by a pure-categorial act, the x and y are indeterminates (variables) which can represent any number whatsoever. Yet, when we see it, we instantly and immediately recognize that it is universally and necessarily valid. We don’t have to go through a tedious process of substituting x and y with different numbers to check if there is some arithmetical exception to this rule. No! Not at all! We know that there are no exceptions to this mathematical truth.
The same happens with some material concepts such as the essential relationships between lines, points, spaces we can see in traditional euclidean geometry. The same we can see in part-whole relations too.
Hierarchy of Objectualities
Now that we understand the three intuitions and mathematical knowledge, we are able to understand something regarding objectualities or states-of-affairs: that there can be hierarchies of objectualities or states-of-affairs.
Let’s say that I have these sets of books:
- Set of Books on Quantum Physics
- Set of Books on Cosmology
- Set of Books on Zoology
- Set of Books on Cells
- Set of Books on Plants
Each of these sets is given to me thanks to the fact that there are books on each of these subjects. These books are the sensible objects, and I am able to constitute them as sets of books by means of objectual acts.
Yet through other objectual acts, I can still constitute a higher-level of sets! For example:
- Set of Books on Physics (Sets of Quantum Physics and of Cosmology Books)
- Set of Books on Biology (Sets of Zoology, Cells, and Plants Books)
Through another objectual act, I can even constitute a higher objectualities:
- Set of Science Books
And I could continue indefinitely! If I wish to find out the sensible objects which are the reference basis for this hierarchy of objectualities or states-of-affairs, all I have to do is to trace down the different objectual levels to the sensible components.
If you want a hierarchy of pure objectualities, just get rid of the sensible components, substituting them by indeterminates, and you have a hierarchy of sets in its purity.
Husserl used sets to illustrate this because it is the easiest example, but this hierarchy is not limited to sets, it also extend to all other formal-objectual categories.
For more technical details on Husserl’s epistemology of mathematics, see Hill & Rosado (2000, pp. 221-239). The original information appears in Logical Investigations, second volume, Sixth Investigation, §§ 40-52, 59-66. In Experience and Judgment he also talks about the hierarchy of objectualities using sets as examples.
How to Solve Some Paradoxes
Do you remember the paradoxes of naïve set theory? Well … Husserl solved them in his philosophy.
Solution to Cantor’s Paradox: In principle, every set can be an element of a higher-order set. This itself blocks the possibility of a "set of all sets". It simply can’t happen in this mathematical scenario.
Solution to the Zermelo-Russell Paradox: The hierarchy of objectualities or states-of-affairs blocks the possibility of any set forming part of itself. In this scenario, no set can form part of itself.
If you wish more technical details on this subject see Hill & Rosado (2000, pp. 235-236).
The First Platonist Epistemology
For all practical purposes, Husserl proposed the first platonist epistemology of mathematics. Many others have really tried to do something similar and failed miserably. Frege, for instance, talked very loosely about "grasping" mathematical objects. Yet, his semantic doctrine reduced numbers to his notion of objects (saturated entities). Husserl conceived numbers as formal structures, making it possible to develop an epistemology of mathematics. Some other philosophers, like James Robert Brown, have talked about an exotic faculty of knowing numbers they call the "mind’s eye", but when examined, it really does not explain anything.
I wish to say, though, that Jerrold Katz tried to develop a similar epistemology in his book Realistic Rationalism, which is (in my judgment) a philosophical gem. For him, mathematical objects are also structures given along sensible objects, and our mathematical intuition consists of getting rid of those sensible objects. However, the difference between Husserl and Katz is two-fold. For Katz, mathematical intuition includes what Husserl would call eidetic intuition. Also, Katz’s theory, unfortunately, is not as clear and sophisticated as Husserl’s.
Finally, I want to point out that Husserl’s criticism to psychologism, empiricism, and naturalism, led him to enrich the current understanding of intuition. These three doctrines tended to limit our knowledge and intuition to sensible intuition, and sometimes they were limited by phenomenalism. Yet, Husserl’s platonist epistemology is powerful precisely because he can also posit categorial intuition and eidetic intuition. These are non-mystical qualities of understanding which we use in every-day life, and are the basis for our mathematical knowledge at every level.
Naturalists today are still essence-phobic. Some of that phobia is justified, but not when it comes to formal sciences. Even eminent minds like W. V. O. Quine criticizes the platonist understanding of meanings, saying that "meanings are what essences become when they are detached from the object and wedded to the word". Yet, his epistemology limits itself to the positing of mathematics and logic as indispensable to science itself, yet, he has some problems:
- He is unable to explain why logic and mathematics are necessary for science. He just argues, from the pragmatic point of view, that they work.
- He is unable to explain the success of the predictions made by logic and mathematics, especially in developments which initially seem pointless: as it happened with non-euclidean geometry, negative roots, Hilbertian spaces, among other mathematical developments.
Paul Benacerraf, in his famous essay "Mathematical Truth", states that for a mathematical proposal to be acceptable, it must account for the objectivity of mathematical truths (something which platonism does very well), and the knowledge of mathematical concepts (which he felt platonism couldn’t accomplish). Husserlian platonism fulfills both requirements very, very well.
Like the Bible says: "By their fruits, ye shall know them."
In essence, what Husserl gives us is the reason we can call this a genuine platonist epistemology. We know formal structures from our experience when we constitute states-of-affairs, yet, when we carry out formalization (categorial abstraction), there is no trace of sensible objects anywhere, and we are also able to intuit the necessary and universal relations among these mathematical objects.
References
Benacerraf, P. (1983). Mathematical truth. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics. (pp. 403-420). UK: Cambridge University Press.
Brown, J. R. (1999). Philosophy of mathematics: an introdution to the world of proos and pictures. London & NY: Routledge.
Hill, C. O., & Rosado, G. E. (2000). Husserl or Frege? Meaning, objectivity and mathematics. IL: Open Court.
Husserl, E. (1973). Experience and judgment. (J. S. Churchill & K. Ameriks, Trans.). London: Routledge & Kegan Paul. (Original work published in 1939).
Husserl, E. (1998). Ideas pertaining to a pure phenomenology and to a phenomenological philosophy. The Hague: Kluwer Academic Publishers. (Originally published in 1913).
Katz, J. (1998). Realistic Rationalism. Cambridge: The MIT Press.
Husserl, E. (2001). Logical investigations. (Vols. 1-2). (J. N. Findlay , Trans.) NY: Humanities Press. (Original work published 1900/1901, 2nd ed. 1913).
Sense and Referent of Propositions (Judgments)
One of the most widely discussed issues in semantics has to do with the sense (meaning) and referent of judgments.
The best known initial proposal was Frege’s. For Frege, two assertive sentences can express two different propositions (he called them "thoughts"), which could refer to one sole object. For instance, if I have two assertive sentences like this:
- The morning star is a planet.
- The evening star is a planet.
then both of these sentences express different senses (different propositions), since "the morning star" and "the evening star" are two proper names which express two different senses, but refer to one sole object. And what is the referent of an assertive sentence? For Frege is its truth-value. Sentences (1) and (2) share the same truth value, therefore they are two senses which designate the same referent: truth.
Yet, no sooner has he proposed this semantic doctrine that we can come up with cases which show a very, very big hole in Fregean semantics. Let’s say that I can come up with two sentences like this:
- The morning star is a planet.
- Paris is France’s capital.
Intuitively we would feel very uncomfortable with the idea that both of these sentences have the same referent, since they talk about two different "facts" (to say it loosely), yet they refer to the same object: truth. ~ Scratching my head ~
For this reason, Frege’s semantics was not very popular, but it fit his own agenda of trying to prove that arithmetic can be derived from logic. Unfortunately he failed. Still, he held more or less this view until 1919, with his famous essay "Thought: a Logical Inquiry". By the way, I love this particular essay, it is of his best, and in terms of train of thought and arguments, it is beautiful … but that’s just me.
As I have said in an earlier post, Husserl did agree with Frege about the sense and referent of proper names. He started to disagree with Frege regarding universal names, since Husserl believed that their sense are concepts and their referents are the objects which fall under these concepts.
Yet, their differences couldn’t be more when it came to assertive sentences.
States-of-Affairs
Husserl was not just a logician, mathematician or semanticist, he was also worried about how to develop an adequate theory of knowledge, searching for principles to explain how did we know stuff. Remember, he was also Franz Brentano’s disciple!
Yet, many people seem to forget that it was in the Philosophy of Arithmetic where he said that in some way we have an experience of sets or groups. Husserl’s change of mind indicated that he no longer wanted to reduce experience to pure sets, but he wanted to recognize all sorts of abstract formal categories as part of those experiences. These categories include:
- Sets
- Relations
- Part-Whole
- Cardinal Numbers
- Ordinal Numbers
So, for him, it were not merely sets, but also there were other ways sensible objects are formally organized.
See … in every experience we have about the world, there are two very different elements we must distinguish:
- Sensible Objects: pencils, houses, notebooks, computers, and so on.
- Formal Components: ways of relating these sensible objects objectively.
Let me give you a particular example of how this is so. Tell me: What do we have here?
You might say: "Pencils". Nah! Look again. You just don’t have "pencils" here. You may constitute them in any the following ways:
- A Set of Pencils
- Three Pencils
- First, Second, Third Pencils
- Pencils in a Row
- One Red Pencil, and other Two Pencils
- One Red Pencil, One Green, One Blue
- …. umm… etc.!
No one constitutes just objects without any pure formal context. If you are watching your computer screen, you are aware that the screen is in front of you, that the glass of water is on the table, that your keyboard is on the desk, that the glass of water is beside the keyboard (it which case … be careful not to drop it on the keyboard :-S), that the pizza is under the mattress (well … this just applies to Oscar Madison in The Odd Couple, and two or three people I know ) …
Husserl calls these sensible objects along with their formal components, states-of-affairs (Sachverhalte).
Formal-Ontological Categories
Husserl’s term for these formal components constituted along sensible objects is formal-objectual categories or formal-ontological categories. They are objectual, because they are constituted at once along with sensible objects. In fact, these formal-objectual categories are founded on these sensible objects in a very real sense. The sensible objects (pens, pencils, computers, etc.), the objects we actually perceive with our senses, or which we can constitute in our imagination (such as an imaginary desk, or pegasus, or fairies), can serve as reference basis for multiple states-of-affairs. Yet, these formal-objectual categories cannot be reduced to sensible objects themselves. Sensible objects are intuited with our senses. Formal categories are intuited with our understanding.
To give you an idea of what I’m talking about. See the examples of the pencils up there? Each one of those bullets derived from our experience of the pencils (a set of pencils, three pencils, etc.) is a state-of-affairs. The differences between states-of-affairs consist of their differences between formal components, even when the sensible objects do not vary at all. The difference between "a set of pencils" and "three pencils" is precisely that one of them is constituted as a set and another as an amount (number). Yet, they retain the same reference basis for those formal components (the pencils as sensible objects). Husserl calls the sensible reference basis situation-of-affairs (Sachlage).
I think that the following example will show clearer manner what Husserl meant. Let’s say that I go to a club and meet two girls (… well, I don’t visit clubs at all … but just for the sake of the argument … ), and they look like this:
Meet Megan (left) and Mary (right). One thing that becomes evident to me is that they are both beautiful and attractive, to the point that I forget everything about philosophy . But I’ll try concentrating 
Another thing that really strikes me about meeting these new friends is that Megan is taller than Mary, and that Mary is shorter than Megan. Yet, what I have just referred to are two states-of-affairs, because their formal relations I just established are different. If I am conscious that Megan is taller than Mary, I am constituting one state-of-affairs. And if I notice that Mary is shorter than Megan, I am constituting another state-of-affairs. On a sensible level, I don’t perceive the "taller than" or "smaller than", just Mary and Megan. Yet, despite we don’t perceive these formal relations, they are objectively (and objectually) founded on sensible objects: Megan and Mary. So … according to Husserl … Megan and Mary comprise the situation-of-affairs, which is the reference basis for two states-of-affairs: Megan being taller than Mary; and Mary being shorter than Megan.
For Husserl, "taller" and "shorter" are two relations, hence formal-objectual categories. As we shall see in a future post. These categories are not the result of a reflection (something that Husserl believed in his psychologistic phase), but they are actually and evidentially given at once along with Megan and Mary. You just open your eyes, look at them, and you immediately constitute these states-of-affairs at once.
And why does Husserl call thes formal-objectual categories also formal-ontological categories? Very simple! As we shall see later, in his platonist phase, Husserl grants these categorial forms an ontology, that is, an independent abstract existence as mathematical objects. He also calls them "ontological" because they are the a priori forms of any being whatsoever, which means that anything that exists must be arranged formally all of these ways.
Sense and Referent of Assertive Sentences in Husserl
After explaining the semantic difference between states-of-affairs and situation-of-affairs, we are in a position to understand Husserl’s semantic doctrine regarding assertive sentences, and notice his huge differences with Frege regarding this. As we have said above, Frege proposed a very awkward doctrine of sense and referent of assertive sentences. For him, the referent of any assertive sentence is a truth-value: truth or falsity. In some occasions, some sentences are neither true nor false, but we won’t get into that now.
On the other hand, Husserl’s semantics are drastically different. He was not a logicist, and it was not his task to show that mathematics could be reduced to logic, but rather logic and mathematics are correlates (and we will explain that later), they are sister disciplines, bound together in a mathesis universalis at the highest level. Therefore, he did not sympathize with Frege’s notion of concept as a function, nor did he find a truth-value to be an object referred to by assertive sentences.
For both, Husserl and Frege (at least the Frege of "On Sense and Referent"), two assertive sentences such as "the morning star is a planet" and "the evening star is a planet" express two different propositions, because they propose two different things. Yet, for Frege, their common referent is a truth-value, for Husserl their referent is a state-of-affairs.
For Husserl, truth is not an object, but a relation between a proposition and a state-of-affairs. For him, a proposition is true if a proposition has a state-of-affairs as referent, and it is false if it does not.
Some Fun Facts …
In my book, The Relation between Formal Science and Natural Science, I talked about the scientific validity of Husserl’s own observations. For instance, to be able to survive, animals must not only constitute objects, but also do so in a certain manner if they want to establish the kind of relationship with their object in order to survive.
Some animal species possess some kind of notion of number. At a rudimentary level, they can distinguish concrete quantities (an ability that must be differentiated from the ability to count numbers in abstract). For what of a better term we will call animals’ basic number-recognition the sense of number. . . .
Domesticated animals (for instance, dogs, cats, monkeys, elephants) notice straight away if one item is missing from a small set of familiar objects. In some species, mothers show by their behaviour that they know if they are missing one or more than one of their litter. A sense of number is marginally resent in such reactions. The animal possesses a natural disposition to recognise that a small set seen for a second time has undergone a numerical change.
Some birds have shown that they can be trained to recognise more precise quantities. Goldfinches, when trained to choose between two different piles of seed, usually manage to distinguish successfully between three and one, three and two, four and two, four and three, and six and three.
Even more striking is the untutored ability of nightingales, magpies, and crows to distinguish between concrete sets ranging from one to three or four.
. . .
What we see in domesticated animals is the rudimentary perception of equivalence and non-equivalence between sets, but only in respect of numberically small sets. In goldfinhes, there is something more than just perception of equivalence — there seem sto be a sense of "more than" and "less than". Once trained, these birds seem to have perception of intensity, halfway, between perception of quantity (which requires an ability to numerate beyond a certain point) and a perception of quality. However, it only works for goldfinches when the "moreness" or "lessness" is quite large; the bird will almost always confuse five and four, seven and five, eight and six, ten and six. In other words, goldfinches can recognise differences of intensity if they are large enough, but not otherwise.
Crows have rather greater abilities: they can recognise equivalence and non-equivalence, they have considerable powers of memory, and they can perceive the relative magnitudes of two sets of the same kind separated in time and space. Obviously, crows do not count in the sense that we do, since in the absence of any generalising or abstracting capacity they cannot conceive any "absolute quantity". But they do manage to distinguish concrete quantities. They do therefore seem to have basic number sense. (Ifrah, 2000, pp. 3-4).
More Fun Facts . . .
Not only animals have a number sense (of what Husserl would call more properly "categorial intuition"), but babies do too! Karen wynn has experimented with five-month-old babies and found that they can perform elementary forms of mental arithmetic. Steven Pinkers tells us all about it:
In Wynn’s experiment, the babies were shown a rubber Mickey Mouse doll on a stage until their little eyes wandered. Then a screen came up, and a prancing hand visibly reached out from behind a curtain and placed a second Mickey Mouse behind the screen. When some screen was removed, if there were two Mickey Mouses visible (something the babies had never actually seen), the babies looked for only a few moments. But if there was one doll, the babies were captivated — even though this was exactly the scene that had bored them before the screen was put into place. Wynn also tested a second group of babies, and this time, after the screen came up to obscure a pair of dolls, a hand visibly reached behind the screen and removed one of them. If the screen fell to reveal a single Mickey, the babies looked briefly; if it revealed the old scene with two, the babies had more trouble tearing themselves away. The babies must have been keeping track of how many dolls were behind the screen, updating their counts as dolls were added or subtracted. If the number inexplicably departed from what they expected, they scrutinized the scene, as if searching for some explanation (Pinker, 1994, p. 59; see Wynn, 1992).
Some Other Issues
One of the very big philosophical problems is to determine what the heck "facts" are. Most philosophers agree, against Frege, that the referent of propositions are "facts", not truth-values. In his essay "Thought", Frege had determined that facts are essentially senses, not referents. For him, facts are nothing more than true propositions (or, in his terminology, true "thoughts").
Wittgenstein was inspired by Fregean semantics, but did not buy this. In the Tractatus, he says that the "world" is not made up of objects, but "facts". And what are facts? He says that facts are "Sachverhalte" (states-of-affairs). His notion of "facts" and "states-of-affairs" are pretty close to the way Husserl used these terms. Like Husserl, Wittgenstein would conceive these "states-of-affairs" as atomic logical units.
On the other hand, Karl Popper does agree that the sense of assertive sentences are propositions and that "facts" are their referent, but he seems to conceive facts more in line with Husserl’s notion of situation-of-affairs. For example, see what he said here:
Many different statements or assertions may equally truly describe one and the same fact. For example, if the description "Peter is taller than Paul" is true, then the description "Paul is shorter than Peter is true (Popper, 1994, p. 102; my emphasis).
By making a semantic distinction between states-of-affairs and situations-of-affairs, Husserl seems to have covered all the bases. For him, states-of-affairs are the facts referred to by propositions. At the same time, these states-of-affairs have situations-of-affairs as reference basis.
So, if we were to summarize Husserl’s doctrine of sense (meaning) and referent (objectuality), we would do it this way.
Finally, notice that Husserl doesn’t bow down to phenomenalism (the doctrine that we are actually given are sense-data: gradations of colors, sounds, tastes, etc.) For him, we are given objects in a specific formal arrangement (states-of-affairs), and all knowledge stems from them. For him, sense-data (he calls them hyletic data) are the result of processes of sensible abstraction. They are never primordially or evidently given first hand.
References
Hill, C. O., & Rosado, G. E. (2000). Husserl or Frege? Meaning, objectivity and mathematics. IL: Open Court.
Husserl, E. (1973). Experience and judgment. (J. S. Churchill & K. Ameriks, Trans.). London: Routledge & Kegan Paul. (Original work published in 1939).
Husserl, E. (2001). Logical investigations. (Vols. 1-2). (J. N. Findlay , Trans.) NY: Humanities Press. (Original work published 1900/1901, 2nd ed. 1913).
Ifrah, G. (2000). The universal history of numbers: from prehistory to the invention of the computer. John Wiley & Sons.
Pinker, S. (1994). The language instinct: how the mind creates language. NY: Harper Perennial.
Popper, K. (1994). Knowledge and the body-mind problem: in defence of interaction. London & NY: Routledge.
Wynn, K. (1992). Addition and subtraction in human infants. Nature, 358, 749-750.
One of the things that may confuse some readers of these series is something I said in this blog post. I quoted Husserl saying that he changed his mind, and moved away from psychologism when he read four particular authors:
- G. W. Leibniz
- Bernard Bolzano
- Hermann Lotze
- David Hume
Now … it seems that we have an "ugly duckling" in this group. Leibniz is to be expected, since he developed a metaphysics which allowed Husserl to see the close relationship between logic and mathematics. Bernard Bolzano saw the same thing, and further elaborated a platonist conception of semantics. Lotze also followed that rationalistic tradition even to the point of positing logicism. … But David Hume??!!!! How does Hume fit in all of this?
At face value it doesn’t seem obvious why reading Hume would change Husserl’s mind regarding psychologism. First, Hume was not a rationalist, but an empiricist. If anything, psychologism owes him a great deal. He was the first one to say that what we are given are impressions, never objects themselves. This led to phenomenalism, the doctrine which states that what we are really given are sense-data. Also Hume was the one who was able to propose the skeptical solution to psychological problems, something which Immanuel Kant couldn’t stand much, leading to his writing of the Critique of Pure Reason.
Yet, there are many aspects of Hume which people generally overlook. First of all, Hume did not consider himself a radical skeptic, but moderate. This is shocking after everything we are usually taught about him, but yeah… he is moderate. Second, his skeptical "solution" is not really a solution, but a genuine problem to be solved in philosophy, and Hume saw it that way. He was pessimistic about it, but without a doubt he was not proposing that all philosophical discussions be closed because he saw no real solution to all the problems he deals in his work.
I will propose something in this article: if we want to understand Hume, we have to go back to René Descartes. In fact, I even make a more radical proposal (well, in reality quite modest, not too radical), that Hume was more Cartesian than many people around the world think. And in examining Descartes and Hume, we will find why Hume was an excellent point of departure from psychologism.
René Descartes and his Skeptical Position
René Descartes (1596-1650)
René Descartes is one of those geniuses of Modernity who actually contributed to three fields, mathematics, philosophy, and the physical sciences. He was a devout Catholic, but he was also knowing the fast changes taking place during his time, he was in a very unique position in history.
In universities all throughout Europe, Aristotelian science (and with it the Ptolemaic view of the universe) was being taught as true …. but there were so many things that happened before which people believed to be true and are no longer true. People used to believe that there was no other territory besides Europe, Africa and Asia, … and (alas!), America was discovered! We believed that the Earth was the center of the universe … but no: Nicolas Copernicus, Johannes Kepler, and Galileo Galilei showed conclusively that the sun was actually in the center. In fact, he was very close to this issue because, when Galileo was condemned by the Inquisition, he though he was next! As he eloquently stated in a letter: Descartes’ discoveries and science depended greatly on Galileo’s. These new discoveries, and the new development in physics implied that the whole Aristotelian metaphysical thinking had to be redone in philosophy.
So … Descartes felt that, in some way, we must start from scratch, and develop a new metaphysical doctrine which would serve as the proper foundations for certain knowledge. So, he developed what today is called the Cartesian method. It goes like this:
- Only accept as true those things that are evidently true in a clear and distinct manner.
- Take a complex whole (which in principle is difficult to understand due to the complexity) and divide it in the simplest parts possible, so these parts can be independently, distinctly, and clearly understood.
- Then, relate these parts, step by step, slowly, until you can reach the whole once again. The difference is that this time, you will understand the complexity because you understand the parts and how they interact.
- Check all of the steps and conclusions carefully, so that you don’t miss anything important.
So, when he had the time, he went through a whole series of philosophical reflections or mediations. He talked about these meditations in his Discourse of Method, and his Metaphysical Meditations on First Philosophy. As a methodology, Descartes assumed what is called today as the Cartesian doubt, which consists of placing into doubt anything that would lead you to the slightest doubt or false conclusions. Here are the steps he took:
- Discard completely all of the prejudices which I was taught as a kid, or those which spring from the fact that I was too opinionated, because experience has shown us time and time again that these prejudices and opinions can be misleading.
- I could trust my senses because they perceive the external world. Yet, my senses have deceived me before. So, I should at least discard my perceptions as source of certain, evident, clear and distinct knowledge.
- Even if I distrusted my senses, there are some things that I feel I should not distrust, such as the fact that I’m in this room, writing on my laptop, feeling the wind of the fan on my face. But even in this case, Descartes would ask something like: "How many times haven’t I dreamed that I was in this place, in this room, writing on my laptop and feeling the wind of the fan on my face?" This could be a very vivid dream! So, let’s discard my confidence in even what appears to be a prima facie experience too vivid to be dismissed, because there is always a chance that it is not the case.
- But what about math? Math provides us certain knowledge, right? "2+2=4" will always be true, no matter what. Except … that even with this apparent evidence, there can be hidden problems in our minds in our acceptance of this proposition as true. Let’s imagine that there is an "evil spirit" which is not only making me believe that I’m here in this room writing, but also that he is making me believe that "2+2=4", when in reality "2+2=3". How can I deny this possibility? It could be, you know! So, I should distrust mathematics.
After this point, one feels like scratching one’s head and asking what the f*** is he doing?! And how with such Cartesian doubt will we reach anything?
Descartes, though, assures us that it is all worth-while. After practically placing everything in doubt … there is still another thing which cannot be questioned: that I am thinking. What is thinking, according to Descartes? He means thinking, affirming, denying, desiring, loving, hating, being concerned, etc. … in other words, any activity of the mind. Even if I wanted to deny that I’m thinking, I am thinking. And if there is a thinking process, then that means that there must be a subject carrying out that thinking process … ME! This thing is the origin of his famous phrase: "I think, therefore I exist" (Cogito, ergo sum).
This truth is an evident, clear and distinct basis to build an entire theory of knowledge which can restore confidence in the world. I won’t discuss how he does that, but for now, we have said enough.
Note: The phrase "I think, therefore I exist" should not be understood as meaning that Paris Hilton, Sarah Palin and George W. Bush don’t exist…. OK? I wanted to clarify that, just in case.
David Hume
David Hume (1711-1776)
David Hume was a fine mind in his time, and perhaps there is no other genuine philosopher who has provoked so many philosophical discussions as he has. We could say that almost everything in philosophy from then on is, in one way or another, a response to Hume.
Once again, he is regarded as an extreme skeptic, formulating a very unsustainable solution to a series of problems he formulates in his books. Yet, again, he describes himself as "moderate". How can this be? If you look at his criticism of Descartes, you’ll understand exactly what he is talking about.
There is a species of scepticism, antecedent to all study and philosophy, which is much inculcated by Des Cartes and others, as a sovereign preservative against error and precipitate judgment. It recommends an universal doubt, not only of all our former opinions and principles, but also of our very faculties; of whose veracity, say they, we must assure ourselves, by a chain of reasoning, deduced from some original principle, which cannot possibly be fallacious or deceitful. But neither is there any such original principle, which has a prerogative above others, that are self-evident and convincing: or if there were, could we advance a step beyond it, but by the use of those faculties, of which we are supposed to be already diffident. The Cartesian doubt, therefore, were it ever possible to be attained by any human creature (as it plainly is not) would be entirely incurable; and no reasoning could ever bring us to a state of assurance and conviction upon any subject (Hume, 1777/1975, pp. 149-150).
In this case, Hume is absolutely right. If we doubt our opinions, our senses, but also our faculty of reasoning through the posit of an "evil spirit", it doesn’t matter what you say or do after that, you cannot build any certain knowledge out of it! Mathematics is based on the faculty of reasoning, and the "evil spirit" conjecture, practially destroys this faculty. You cannot state with assurance that "I think, therefore I exist" is evident … what prevents us from believing that the "evil spirit" is making us think that?
So, for Hume, the Cartesian doubt is an extreme form of skepticism. Yet, contrary to what people think, he did not throw Descartes’ works to the waste basket, and this is the evidence (and I’ll highlight it):
It must, however, be confessed, that this species of scepticism, when more moderate, may be understood in a very reasonable sense, and is a necessary preparative to the study of philosophy, by preserving a proper impartiality in our judgments, and weaning our mind from all those prejudices, which we may have imbibed from education or rash opinion. To begin with clear and self-evident principles, to advance by timorous and sure steps, to review frequently our conclusions, and examine accurately all their consequences; though by these means we shall make both a slow and a short progress in our systems; are the only methods, by which we can ever hope to reach truth, and attain proper stability and certainty in our determinations (Hume 1777/1975, p. 150).
See?! In terms of methodology, Hume is Cartesian. He adopts the Cartesian method as his own!
But then the question is, where does Hume show "moderation" in his philosophy? Look again at his criticisms to Descartes. One big problem he had was that he placed into question our own faculties of reasoning. So, the solution to this problem (if we want to go anywhere philosophically) is not to place into doubt our basic logical faculties.
Hume had read G. W. Leibniz who established a difference between truths-of-reason (vérités-de-raison) and truths-of-fact (vérités-de-fait). For Leibniz, truths-of-reason are those which can be reduced to logical principles alone, and which are always necessary (in the sense that it is necessary that "2+2" be "4"), and are true in every possible world. On the other hand, truths-of-fact are statements whose truth is not self-evident, and they are contingent (i.e., not necessary).
David Hume established something close to that distinction. He made the difference between relations-of-ideas and matters-of-fact. He defines "relations-of-ideas" this way:
[Relations-of-Ideas] arethe sciences of Geometry, Algebra and Arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain. That the square of the hypothenuse is equal to the square of the two sides, is a proposition which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Proportions of this kind are discoverable by mere operations of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence (Hume, 1777/1975, p. 25).
Today we would clarify that as long as we are talking about euclidean space, this is obviously true. But what are matters-of-fact?
Matters of fact … are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality. That the sun will not rise to-morrow is not less intelligible a proposition, and implies no more contradiction, than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood. (Hume, 1777/1975, p. 25-26).
In other words the negation of any statement belonging to relations-of-ideas should lead to a contradiction. To deny that "2+2=4" would imply necessarily a contradiction. This is not the case of the matters-of-fact. The fact that the sun rose today does not mean that it will rise again tomorrow. The fact that it may not rise tomorrow is not logically self-contradictory, therefore it is a possibility. Here is the basis for Hume’s criticism of induction … which we will discuss another time.
Husserl’s Lesson from David Hume
Why did reading Hume help Husserl change his mind? Let’s not be confused, Husserl did recognize that Hume’s skepticism was extreme (Prol. § 51), even when it was far more moderate than psychologists like John Stuart Mill (Prol. § 26 Ap.) Hume falls in psychologism, but still, he is more prudent.
The reason for Husserl’s change of mind reading Hume can be found in his Logical Investigations.
We are here concerned with the repeatedly stressed distinctions between real and ideal truths, laws, sciences, between real and ideal (individual or specific) generalities and also singularities, etc. Everyone, no doubt, has some acquaintance with these distinctions: even so extreme an empiricist as Hume draws a fundamental distinction between "relations of ideas" and "matters of fact", a distinction which the great idealist Leibniz drew before him, using the rubrics vérités de raison and vérités de fait (Prol. § 51).
In other words, both Leibniz and Hume, from very different points of view, did actually distinguish between two very different realms of truth, ones which are necessarily true (relations-of-ideas), and one which aren’t (matters-of-fact). Husserl would then distinguish between the ideal, meaning the atemporal realm of meanings, logical relations, and mathematical objects in general; and the real, where we find the temporal realm where physical objects and mental activities take place. Propositions about ideal relations and objects are for Husserl "relations-of-ideas", which would include all analytic-a priori and synthetic-a priori propositions, while propositions about the sensible world or our psychological activity would be for him "matters-of-fact". These propositions would all be synthetic-a posteriori.
Even though Husserl had his serious reserves regarding David Hume, he would use his terminology (relations-of-ideas and matters-of-fact) throughout the years, especially in most of his key logical and phenomenological works.
References
Hume, D. (1975). Enquiries concerning human understanding and concerning the principles of morals. L. A. Selby-Bigge & P. H. Nidditch (eds.). Oxford: Clarendon Press. (Original work published in 1777).
Husserl, E. (2001). Logical investigations. (2 vols.) London & NY: Routledge.
Introduction
One of the things I’ve learned since I began studying philosophy is that our theoretical framework determines what we observe, what we see. One of the examples I learned this from was in a very good anthology of readings on Gottlob Frege, edited by Michael Beaney. I highly recommend it, Beaney does a great job in compiling essential readings of Frege, as well as some translations made by Beaney himself.
The only thing I don’t like about it is his refusal to translate the word Bedeutung. Beaney’s attitude is understandable, Frege made a very poor choice of words regarding that term, since the word Bedeutung in German usually means "meaning" (pardoning the redundancy), while Frege uses it to mean "referent" or "denotation". In Logical Investigations, Husserl quotes Frege only two times in all of that magna opus, and in one of those occasions was precisely to criticize Frege’s use of the word Bedeutung in this particular way (see Inv. I, § 15). So, I understand Beaney’s refusal to keep the German word Bedeutung, but if by that, Frege means "referent" … then I think it should be translated as "referent".
The book has the received view originating from an essay by Dagfinn Føllesdal, a renowned philosopher and W. V. O. Quine’s disciple. This essay was called "Husserl’s Notion of Noema" published in the Journal of Philosophy in 1969. In it, Føllesdal states that Husserl converted to platonism and left psychologism as a direct result of Frege’s review, something which, by this stage of our series we know to be false. That is not the only allegation, though. He also says that when we look at Husserl’s notion of "noema" in his phenomenological stage, we must realize that this notion is nothing more than a generalized version of Frege’s "senses" (Sinn). Husserl made the distinction between sense and referent in Logical Investigations, which was written in 1900, clearly after Frege’s review in 1894, and Frege’s famous masterpiece "On Sense and Referent" (1892). And for Husserl, a noema is nothing more than an a sense (meaning) given by an intentional act. Hence, a noema is the sense and the referent is an object. Right? So, Husserl apparently became a platonist a la Frege and his contribution is the generalization Frege’s senses to the realm of presentations. Without Frege, no phenomenology (at least a la Husserl). Interesting!
As I said, Beaney’s book apparently subscribes to this view originated by Føllesdal’s essay (Beaney, 1997, p. 6). Part of what makes Beaney’s anthology so good, though, is that he includes in it a portion of Frege’s letter to Husserl dated May 25, 1891. This letter was a response to Husserl sending Frege his Philosophy of Arithmetic, and his review on E. Schröder’s work on mathematics. In it, Frege says the following (I’ll translate "Bedeutung" as "referent"):
Dear Doctor,
…
I thank you especially for your Philosophy of Arithmetic, in which you take notice of my own similar endeavours, perhaps more thoroughly than has been done up to now. I hope to find some time soon to reply to your objections. All I should like to say about it now is that there seems to be a difference of opinion between us on how a concept-word (common name) is related to objects. The following schema should make my view clear:
With a concept word it takes one more step to reach the object than with a proper name, and the last step may be missing — i.e., the concept may be empty — without the concept word’s ceasing to be scientifically useful. I have drawn the last step from concept to object horizontally in order to indicate that it takes place on the same level, that objects and concepts have the same objectivity. . . . Now it seems to me that for you the schema would look like this:
so that for you it would take the same number of steps to get from proper names to objects as from concept-words. The only difference between proper names and concept-words would then be that the former could refer to only one object and the latter to more than one. (Beaney, 1997, pp. 149-150).
SEE???!!!! Right THERE!!!!! … Beaney apparently didn’t see it in his own anthology, but it’s there!!! Get it???!!!!
If you didn’t get it, let me spell it out for you. Føllesdal argued in 1969 that Husserl made the difference between sense and referent as a result of Frege’s essay "On Sense and Referent" (1892), and Frege’s review (1894). Yet, here, in a letter written in 1891(!!!!), Frege says clearly, that Husserl had made the distinction between sense and referent by that time. Both men seem to have come up with the distinction of sense and referent simultaneously, but independently … much like the way Newton and Leibniz both developed calculus.
This distinction did not appear in Frege before 1890, but this distinction did appear in Husserl’s review on Schröder’s work, a review published in 1891. Føllesdal was wrong, once again. J. N. Mohanty, Claire Ortiz Hill, and Guillermo E. Rosado Haddock have worked extensively on this subject of Husserl’s development of his semantic doctrine. Claire Hill (2001) and Jaakko Hintikka (1995) have worked also in clarifying the relationship between Husserl’s notion of noema, and Frege’s notion of sense. Below I give you the references if you wish to look for them.
For now, I wish to ask the question. How come two very different philosophers come up with almost the same distinction? The answer lies in the mathematical and semantic studies of a relatively unknown Czech priest at the time called Bernard Bolzano, whose work Husserl and Frege knew very well. I’ve said in a previous blog post that Husserl’s reading of Bolzano’s works helped him turn away from psychologism. Well … here is a very brief version of his story and why he influenced Husserl the way he did.
A Priest and the Birth of Semantics
Born in Prague in 1781, Bernard Bolzano studied mathematics, philosophy, and physics in the University of Prague. Due to his dedication and devotion to God and the Catholic Church, he studied theology and was ordained a priest in 1804, and later appointed to be a position in philosophy of religion. Due to political problems and his stance for peace, he was dismissed from the university in 1819, and he spent the rest of his life writing on many subjects, including mathematical and philosophical. He never had too much exposure mostly because he was forbidden to publish in mainstream journals as a condition for his exile. He had no choice but to publicize his ideas in other sorts of journals and publishing businesses, not as widely circulated. He died much later in 1842.
Despite the obscurity of such publishing methods, his writings reached the hands of Franz Brentano, Edmund Husserl, Georg Cantor, and others, such as Gottlob Frege. In fact, Husserl first heard of Bolzano in one of Brentano’s seminars. He read him, and although very impressed by his philosophy, he felt pretty much uncomfortable with some "objective, existent, abstract" beings posited by Bolzano, which for Brentano were nothing more than fictions. As a mathematician and a proponent of psychologism (that is, before 1890), Husserl found it difficult to deal with Bolzano’s philosophy.
One of the most important works by Bolzano was Theory of Science, where he pondered a lot about some issues which seemed to arise from mathematics. One of them has to do with the core issue of all philosophy of mathematics: "what the he** is a number?" One thing is to represent a number this way "2", or this way "II" or this way "two". These different representations are obviously just signs, i.e. physical expressions which stand for something else. Yet, numbers don’t seem to be physical objects either, because in mathematics we never really deal with propositions like "two oranges and two other oranges make up four oranges", but "2+2=4". We deal with numbers in their purity from all physical content. How do we deal with this?
Bolzano distinguished between what he called "subjective representations" and "objective representations" (if this sounds like the Husserl of Philosophy of Arithmetic, it is no accident). Many people can have a subjective image or representation of their own minds, yet it seems as if despite the differences in our representations, there is a general agreement of something objective we do seem to share. If I show you a picture of the phone, each one of you will see it in a different manner, because sight depends on perspectives, where the light hits, if you are color-blind, and so on. Yet we all understand what a phone is in an objective manner despite our different mental ssubjective representations.
In the same way, we can represent numbers with different physical signs, either through symbols in writing or the sound when we pronounce the words for the number two in different languages ("two", "dos", "deux", "zwei", etc.) Yet, despite these psychological or physical differences with the way we represent ourselves the number two, there is a sort of objectivity we share. That is the objective representation we somehow understand (but do not represent in our own imagination) in our minds.
Bolzano says something interesting:
[Objective representations] are not to be found in the realm of the real. An objective representation does not require a subject but subsists, not indeed as something existing, but as a certain something even though no thinking being may have it; also, it is not multiplied when it is thought by one, two, three, or more beings. . . . Fort his reason, any word, unless it is ambiguous, designates only one objective representation (Coffa, 1991, p. 30).
By not existing, what Bolzano meant is that "objective representations" are not in space or time as all physical objects are, yet subsist in some way or manner. They are different from mental representations, and at the same time the object (referent) of that representation.
He also talked about propositions, that is, those sentences which can be said to be true or false: "The grass is green", "The car is yellow", "The computer screen is not working". There are subjective representations of these propositions, the physical ink and the particular letters, or whenever you hear the sounds when I say these things.
Yet, there is also an objective side of propositions, these he would call "propositions-in-themselves" (Sätze an sich), which is made up of objective representations and they are related through copulae (and, or, if … then, and so on). So, if I say that "the car is yellow", the words "car" and "yellow" express an objective representation of both a car and the color yellow, but joined together with the word "is". With the word "is", we have linked the two (the car and the color yellow) in a subject-predicate manner. In essence, for all practical purposes, Bolzano is developing a theory of grammar, so that propositions make sense (or have meaning). This is precisely what semantics is all about (theory of meaning).
Unfortunately Bolzano nor Frege or Husserl used the word "semantics" to describe it. It was still called "logic" at the time. But, one thing that this theory led is to a doctrine of analytic and synthetic judgments. Coffa (1991) sums up very well Bolzano’s point using two examples:
- This man is a featherless biped.
- If all men are mortal and all Greeks are men, then all Greeks are mortal.
The difference between both propositions is that the first one is a posteriori (which means that it has to be verified by experience). The truth of proposition (1) will inherently depend on whether that particular man is a featherless biped.
We cannot say the same thing about proposition (2). It seems as if the truth of this proposition did not actually depend on the existence of men, or Greeks, or even mortal beings. The absolute truth depends on the grammatical arrangement, or the form of the proposition. To give you an example:
"If all teavies are born in Paris, and if Paris is Honduras’ capital, then all teavies are born in Hondura’s capital".
And this is absolutely true despite the fact that no one knows what "teavies" are, and that Paris is not Honduras capital. The grammatical structure "if … and if … then …" seems to make all the difference. Eventually, Husserl will adopt this view to elaborate his own doctrine on analytic and synthetic judgments.
Husserl and Bolzano
Husserl considers his views as being a more sophisticaded and elaborated version of Bolzano’s ideas. Bolzano’s use of the term "representations" (Vorstellungen) which is a psychological term, to refer to two different things: one subjective, the other objective. Husserl’s error when he was a psychologist, was to believe that these objective representations are in some sense psychological in origin. Bolzano, clearly, did not think that way, and he was right.
Despite Bolzanos genius, his use of such a language did not let him reach the final difference between sense and referent. One is a bit disappointed, because it is watching a tree whose fruits are all ripe and mature, and all you have to do is to shake the tree a bit so that the fruits finally fall. Husserl and Frege, independently, were able to finally make the distinction, which was hidden in Bolzano’s misguided use of the term "objective representation" to refer to nothing more than sense or meaning.
As we shall see here, as in the case of Bolzano’s "objective representations", senses or meanings are themselves totally abstract but objective. They are not found in space or time, but in a sense subsist independently of the mind or the physical world. They are not the physical sign, nor do they belong in the psychological activity of the brain. If I say: "Kennedy was killed in 1963", its meaning, its proposition, is true, and will always be true, even if everyone believed it to be false. There is no better example to show that senses or meanings are independent of the activities of the mind. Psychologism’s error was to believe that they were not independent. As we shall see in a future blog post, this will lead to inevitable contradictions.
Husserl semantic doctrine, a doctrine of meaning is indeed exactly as Frege described in his 1891 letter, and Husserl will restate his semantic doctrine in his Logical Investigations (Inv. I. § 12).
Sense (Meaning) and Referent of Proper Names
For Husserl, there are two sorts of names: a proper name or a universal name. For Husserl (as for Frege), a proper name is a word or phrase which is used to refer to one single object. It could be a strict proper name such as "Aristotle", "Lisa DeBenedictis" or "Martin Luther King", or what Bertrand Russell would call "definite descriptions" or "denoting phrases" such as: "the king of France".
Let us use the following examples to illustrate the difference between proper names, their sense (meaning), and their referent (I’m going to use a mix of Frege’s and Husserl’s examples):
Example 1:
"The victor in Jena"
"The victor in Jena"
Both proper names are exactly the same, because we are using the same sign or physical represenation in graphic writing. Hence, they contain the same message (sense or meaning), and refer to one and the same object: Napoleon Bonaparte.
Example 2:
"The victor in Jena"
"El vencedor en Jena"
Here both proper names differ, because the graphical and written representations or signs are different. One of the phrases is in English, the other one is in Spanish. Yet they both carry the same sense or meaning, because they tell us the same thing in both languages. They still refer to the same object: Napoleon Bonaparte.
Example 3:
"The victor in Jena"
"The defeated in Waterloo"
Here, both proper names differ, but there is now a difference in sense or meaning. They both refer to Napoleon Bonaparte, but not in the same way. Why? Because both of these proper names are giving us two very different informations about the guy. Since their logical content is different, their sense or meaning is different.
The same is true when we have other sorts of proper names which give us different abstract content (sense or meaning) but referring to the same object: "the equilateral triangle" and "the equiangular triangle"; or, to use Frege’s own example, "the morning star" and "the evening star" to refer to planet Venus.
Example 4:
"The morning star"
"The defeated in Waterloo"
Both proper names are different, both senses or meanings are different, and they both refer to two different objects: the first refers to Venus, and the other one Napoleon.
So, we can sum up these examples in the following way in this table:
Sense (Meaning) and Referent of Universal Names
The theory of sense (meaning) and referent of universal names is exactly as Frege described above. For Husserl, a universal name is whatever names a set of objects. For example, the term "horse" is a universal name. Its sense or meaning is a concept, while the referent are all of those objects which fall under that concept (or extension of the concept).
For example, the universal name "horse" expresses a concept which can have a variety of objects: Black Beauty, Rocinante, Napoleon’s white horse, and all of the horses which have existed and exist today. This works out pretty well in today’s semantics, and it has been adopted in contemporary philosophy.
On the other hand, Frege’s version was left behind. Because of his logicist proposal, and his notion of the concept as a logical function of one argument, he had to place the concept itself at the level of referent (as he graphically explained in Husserl’s letter). Yet, as Guillermo Rosado Haddock has pointed out, in his many of later works, Frege never clarifies what he means or how to exemplify the sense of a concept-word, hence leaving a big hole in his semantic doctrine.
———————————-
But what about "propositions"???!!! We will talk about it in our next blog post.
References
Bernet, R., Kern, I., & Marbach, E. (1999). An introduction to Husserlian phenomenology. IL: Northwestern University Press.
Beaney, M (ed.). (1997). The Frege reader. US: Blackwell.
Coffa, J. A. (1991). The semantic tradition from Kant to Carnap: to the Vienna Station. US: Cambridge University Press.
Føllesdal, D. (1969). Husserl’s notion of noema. Journal of Philosophy, 66, 680-687.
Hill, C. O. (2001). Word and object in Husserl, Frege, and Russell: the roots of twentieth century philosophy. US: Ohio University Press.
Hill, C. O. & Rosado Haddock, G. E. (2000). Husserl or Frege? Meaning, objectivity, and mathematics. US: Open Court.
Hintikka, J. (1995). The phenomenological dimension. In B. Smith & D. W. Smith, The Cambridge companion to Husserl. US: Cambridge University Press.
Husserl, E. (2001). Logical investigations. (2 vols.) London & NY: Routledge.
Mohanty, J. N. (1974). Husserl and Frege: a new look at their relationship. Research in Phenomenology, 4, 51-62
Mohanty, J. N. (1982a). Edmund Husserl’s theory of meaning. The Hague: Martinus Nijhoff.
Mohanty, J. N. (1982b). Husserl and Frege. IN: Indiana University Press.
Rosado Haddock, G. E. (2006). A critical introduction to the philosophy of Gottlob Frege. US: Ashgate.
Introduction
It is 1892, and Husserl was going to adopt a very unusual position in philosophy … one not very popular among philosophers. In fact, not even popular among mathematicians. He had to be honest, you know … but he had mixed feelings about it. Husserl’s change of mind in the start of the 1890s implied that he would have to be true to a dear teacher and friend, Franz Brentano, and tell him about how useless his view of geometry is.
Psychologism went through a Kantian path regarding geometry. For Immanuel Kant, there is an intuition of space which has four essential traits:
- Kant conceived space as a form of intuition, an a priori condition for objects appearing to us as three-dimensional.
- He conceived space the Newtonian way (as opposed to the Leibnizian). Leibniz’s conception of space is relational; for space to exist, there must be objects which can be related spacially or occupy space … no objects, no space. Newtonian conception of space on the other hand is absolute, existing with independence of objects themselves.
- He conceived space as Euclidean, and that is the only way it can be given in experience.
- Geometry as pure mathematics (mathemata) can be constructed in pure intuition: as he called it "constructions from concepts".
Psychologism thought geometry as being the result of abstraction from experience, which is intuitively a three-dimensional euclidean space.
Yet, in his letter to Brentano in 1892, Husserl expressed him his change of mind. No longer was pure geometry to be considered a mere abstraction from experience itself, no longer about generalizations of objects of experience. Geometry is a science in its own right, ruled by all sorts of formal laws. And not only that … Husserl recognized that there was no a priori reason to restrict ourselves to euclidean geometry. Non-euclidean geometry is just as legitimate in all of its forms. Euclidean space is just one of an infinity of possible spaces.
This is indeed not a popular position.
What is Euclidean Geometry?
Simply speaking, euclidean geometry is the sort of geometry developed by Euclid, the famous ancient mathematician. It considers three-dimensional space as perfectly "flat" so-to-speak. If we translate three dimensions into two dimensions, we could represent space as a flat surface.
The geometry you learned in high school is precisely such geometry. In this sort of space, the angles of squares are all right angles, the Pythagorean Theorem applies to all right triangles. In this kind of space, if you add the three angles of a triangle they are equal to 180⁰ (not more and not less).
What is Non-Euclidean Geometry?
It is not obvious to everyone that there can be other sorts of spaces than those we are experiencing intuitively right now, nor those we learned in high school.
The problem from the very beginning had to do with what many mathematicians historically called the axiom of the parallels. It basically states that for any given line a, and a point outside of that line, there can only be one, and only one, line b which is parallel to line a. (Remember, lines extend to infinity, and paralell lines will never intersect at any point of infinite geometrical space).
Interesting, isn’t it? For millenia, this has always been an issue in geometry. At face value, it seems that this is self-evident for many people in geometry. All self-evident truths in mathematics are called axioms, which basically means that they are so extremely evident (almost to the point of exclaiming "duh!") that they do not need any proof at all. Yet, many other mathematicians did not regard this "axiom" of the parallels as self-evident, and stated that there was a need for mathematical proof.
Many people engaged in the search for such a proof throughout history. For instance, during the seventeenth and eighteenth century, a Jesuit priest called Gerolamo Saccheri (1667-1733) used a very important approach widely used in philosophy and mathematics: the use of Reductio ad Absurdum. How does this procedure work? Let’s say that you want to prove that the axiom of the parallels is necessarily and universally true. To do it, you begin by supposing the opposite: we are going to suppose (just for the sake of the argument) that the negation of the axiom of the parallels is true. If in the process of logical deduction from that, it produces a contradiction, then the negation of the axiom of the parallels is regarded as "absurd" (i.e. leads to contradictions), and, therefore, the axiom of the parallels is true.
Saccheri applied this procedure … he supposed the negation of the axiom of the parallels and derived from there using logical and mathematical rules. Alas! He did not find a contradiction. Quite the opposite … he found that the negation of such "axiom" was perfectly consistent. Contrary to what he expected, Saccheri accidentally proved that non-euclidean space is logically possible. He tried to save his views of the inherent and necessity of euclidean space saying that only such a space is intuitive: it is the only real one.
Carl Friedrich Gauss (1777-1855)
Yet, this final judgment by Saccheri did not stop a mathematician called Carl Friedrich Gauss from exploring the mathematical possibilities open by Saccheri’s accidental conclusions. If the axiom of the parallels is not an axiom strictly speaking, then that means that euclidean space is not the only mathematically valid sort of space there is. He wanted to explore other possibilities negating the so-called axiom of the parallels. He discovered that it was perfectly possible that in many ways more lines can pass through point X which are parallel to line a.
But this discovery was further elaborated by János Bolyai. Bolyai’s father, who happens to have been Gauss’ friend, tried to prove the self-contradiction of non-euclidean space, but to no avail. On the other hand, Bolyai (1802-1860) and, simultaneously and separately, Nikolai Lobachevsky (1793-1856) elaborated what is known today as hyperbolic geometry, a set of non-euclidean spaces where space has a particular shape, my like a horse’s saddle.
Yet, there was this other mathematician, also obsessed with non-euclidean geometry called Gerard Bernhard Riemann (1826-1866) who developed another kind of non-euclidean geometry called elliptic geometry. One of the possible elliptical spaces is a spherical-shaped space.
Both of these sorts of spaces are inherently different with respect to euclidean space, and different among each other. Let’s see some of these differences. For instance, in a hyperbolic space, if you have a line a, there can be multiple parallel lines which can pass through any point X, outside line a. In an elliptic space, there can’t be parallel lines, because two straight lines will always intersect, no matter what. In euclidean space, there can only be one parallel line.
Another remarkable difference between spaces is the addition of the angles of a triangle. In Euclidean space, if you add up the angles of a triangle, they will always give you 180⁰; but in hyperbolic space, the result will be less than 180⁰; and in the case of elliptic space, it will always be more than 180⁰.
There is another very interesting aspect to non-euclidean spaces, and it was pointed out by Riemann himself. Imagine, for instance that you are standing on the Earth … like you usually do. Haven’t you noticed that it seems that the Earth is flat? In fact, many people in ancient times did believe that. If you measure one small area like New York City, you are going to confirm that the Earth is flat, but as you increase the size of the measurement to, let’s say, from New York City to the Earth’s equator, you’ll notice that the Earth is no longer flat, but curved.
So, in many small areas of a spherical space, the mathematical truths of euclidean space and those of elliptic space are almost identical, the differences can be negligible for all practical purposes …. just as we know that the Earth is a sphere, but for all practical purposes, we can neglect this fact if we want to measure the area in our backyard. Riemann pointed out that there can be different degrees of measurements this way.
Also, we have to take into account that multi-dimensional euclidean and non-euclidean space can be measured and calculated mathematically. In Riemann’s time, it was thought that physical space was an euclidean three dimensional. Yet, mathematically speaking, four-dimensional space, or five dimensional, or even million-dimensional, or n-dimensional spaces are possible.
All of this means that physical euclidean space as we know it is just one out of many possible spaces. The measurement (metric) of physical space can actually be determined empirically (by experience).
This constitutes Riemann’s notion of manifolds, we can take different "regions" of space and determine their metric properties. Husserl, inspired by Riemannian manifolds, would generalize this notion to the whole of mathematics as a mathesis universalis. In its supreme form, Husserl would consider mathematics as a theory of manifolds, where certain "regions" of mathematics can be determined by positing axiomatic rules (or eliminating some of them), and positing sorts of numbers or mathematical concepts, and derive all sorts of consequences out of them as long as logical consistency is preserved. That is why a mathematician is free to posit fractions, negative numbers, negative roots, decimals … you name it, and out of them derive a completely consistent mathematical theory.
Consequences of the Adoption of Non-Euclidean Geometry
As I said before, Husserl’s position was a philosophical oddity at his time. As far as philosophers in general knew, mathematical obsession with non-euclidean geometry was a waste of time. The vast majority of philosophers (and mathematicians) believed that Saccheri was right: even if non-euclidean spaces were mathematically possible, there was absolutely no reason why we should explore them.
As far as these philosophers go, mathematics is a technique, not a theory itself about anything real. It is merely a means to an end, and it should develop all of those mathematical areas that evidently will be useful for us … we have to discard the rest to the waste basket as some sort of aberration. This was the same position held by many psychologists (i.e. proponents of psychologism) at the time regarding logic. Logic is the art of correct reasoning. Reasoning is a psychological process. Therefore, logic is a psychological technique which should be developed as long as it helps us think clearly, from a psychological standpoint. Forget about treating logic as a theory in itself … or else it will become as useless as non-euclidean geometry. Frege was furious at such attitudes!
After 1890, Husserl discarded all of these opinions to the waste basket. His mathematical side just wouldn’t allow it. Pure logic is not a technique, it is a formal theory which is a pre-condition for any other deductive theory used by science and other fields. Formal logic is a field of its own (not a branch of psychology): in other words, as he would say, "pure logic is a theory of all possible forms of theories" or a "theory of deductive systems". Mathematics was not a technique either, it is a theory of manifolds in its supreme form, all objectualities in the universe have to be understood according to mathematics, not the other way around.
And essentially, that was what Husserl tried to say to his dear teacher and friend, Franz Brentano. Just like Bernard Riemman, János Bolyai, and Nikolai Lobachevsky, Husserl broke Kantian tradition of euclidean space as the only valid space in which we can obtain knowledge.
Ironies of History
Henri Poincaré (Left); Albert Einstein (Right)
Another famous mathematician was Henri Poincaré (1854-1912), today considered by many as one of the fathers of the general theory of relativity. Why would that be? He made bunches of contributions at the time to special relativity, but the big one had to do with the whole idea of non-euclidean geometry. He was not a platonist, but a constructivist. For him, mathematics are about constructions of the human mind.
He realized that non-euclidean geometry was perfectly legitimate, and opened the door to the possibility of a non-euclidean theoretical explanation of the physical world. However, contrary to many philosophers and scientists of the time, he did actually open the door to the idea that experience may determine that a particular non-euclidean space may serve as basis for a better explanation of phenomena in the physical world than euclidean geometry. Unfortunately, his constructivist prejudices led him to believe that this would never happen because "obviously" every-day space is euclidean.
Albert Einstein (1879-1955) did read Henri Poincaré, and had an idea to solve a scientific problem. If space-time itself (space and time as one entity) is physical, not just a pure nothing, and the velocity of any inertial reference frame affects the measurements of mass, distance, and time, then that might suggest a different conception of space and time as they have been traditionally treated in history.
According to special relativity, nothing can travel faster than light’s velocity, something which left a very big hole in Newton’s conception of gravity. According to Newton, the gravitational influence of one mass to another is instantaneous. So, if the sun disappeared all of the sudden, the Earth and all of the other planets would fly away instantly. This is impossible according to special relativity, because the velocity of gravitational effect should have the velocity of light as its limit.
Under these circumstances, Einstein had two choices:
- Either assume that space-time is a very simple euclidean space, while it unnecessarily complicates the scientific theory to explain gravity.
- Or assume that space-time is non-euclidean, which would complicate the geometrical model, but it would simplify considerably the explanation of gravity in light of special relativity.
He chose to assume that space-time is non-euclidean, which helped him suppose that space-time is itself curved by the presence of mass and energy. The reason why you are stuck to the ground (gravitationally speaking) is not because the Earth is "pulling" you towards it as Newton supposed, but rather because the Earth’s mass creates a four-dimensional space-time distortion around it like the following illustration, and you are literally sliding downwards in that space-time. (In the following illustration, the three-dimensional space-time curves are represented as a two dimensional surface around the Earth).
(Illustration courtesy of Johnstone)
The Earth orbits around the Sun because, like a marble over a curved surface, the Earth moves along the curve of space-time created by the massive presence of the sun. If the sun disappears, though, it would create a space-time wave whose gravitational influence would not reach us until about 8 minutes later (the time it takes light to reach us). Einstein, then, was able to provide a better explanation than Isaac Newton for phenomena which Newton accounted for, plus more: he could also explain why light bends in a certain way when it is close to a massive object, the Doppler Effect, and the second twin paradox.
In its attempt to stick to the sensory experience, psychologism was a disaster. Its insistence to stick to euclidean space because that is our empirical experience got the best of it, especially those philosophers who were led by Kantian and Neo-Kantian philosophies. Frege complained that a psychologistic view of logic had actually stalled logic’s development, and he was quite right. If you say that logic is a technique to think well in a psychological level, then any logical discovery which deviates from that does not serve to that end, and would be regarded as useless. Husserl also believed that, although mathematics was far more developed than logic (he called it "logic’s fat sister"), the irruption of psychologism in mathematics would be a transgression to a completely different field. If mathematics is a technique to understand the world, then we should prevent further development in areas that some people at one time would regard as "evidently useless".
Even during the 1980s, antiplatonists such as Phillip Kitcher would be so annoyed by what he considered "useless fat" of mathematics, that he said that we should cut it and throw it to the waste basket.
Imagine what would have happened to science, especially to Einstein and special-relativity, if mathematicians would have actually carried out what these science-loving people suggested.
Final Note: This article serves to refute one of the most widespread prejudices in analytic philosophy since W. V. Quine’s essay "Two Dogmas of Empiricism". For Quine, there is no distinction between formal and natural sciences (analytic or synthetic judgments), because they constitute a unitary whole which can be revised in light of recalcitrant experience; hence, recalcitrant experience can revise logic and mathematics. This conviction was further reinforced by Hilary Putnam’s statements that there have been instances where logic and mathematics were revised because of experience, he mentions the general theory of relativity’s revision of geometry as one of those instances.
If you have been paying attention, you have to realize that mathematical revision took place, but not because of "recalcitrant experience", but because of a problem within the realm of mathematics: the problem of the so-called "axiom of the parallels". Within mathematics, non-euclidean geometry was always mathematically valid even before the success of the general theory of relativity. Einstein never revised non-euclidean geometry, he only adopted it as a mathematical model on which to build his scientific theory, and he succeeded. Similar events have happened in science, for example, Hilbertian spaces have served as basis for many areas in quantum physics, or the way chaos theory and fractals have served as mathematical models to explain many phenomena in the world.
References
Bernet, R., Kern, I., & Marbach, E. (1999). An introduction to Husserlian phenomenology. IL: Northwestern University Press.
Coffa, J. A. (1991). The semantic tradition from Kant to Carnap: to the Vienna station. UK: Cambridge University Press.
Einstein, A. (1983). Sidelights on relativity. US: Dover Publishers.
Frege, G. (1999). The foundations of arithmetic. IL: Northwestern University Press.
Gillies, D. (1993). Philosophy of science in the twentieth century: four central themes. Oxford & Cambridge: Blackwell.
Gleick, J. (1987). Chaos: making a new science. US: Penguin Books.
Gullberg, J. (1997). Mathematics: from the birth of numbers. NY & London: W. W. Norton & Company.
Harrison, E. R. (1981). Cosmology: the science of the universe. Cambridge: Cambridge University Press.
Hill, C. O. (1991). Word and object in Husserl, Frege and Russell. US: Ohio University Press.
Hill, C. O., & Rosado, G. E. (2000). Husserl or Frege? Meaning, objectivity and mathematics. IL: Open Court.
Husserl, E. (1969). Formal and transcendental logic. (D. Cairns, Trans.) The Hague: M. Nijhoff. (Original work published in 1929).
Husserl, E. (2001). Logical investigations. 2 vols. (J. N. Findlay, Trans.). London & NY: Routledge.
Kant, I. (1998). Critique of pure reason. (P. Guyer & A. W. Wood, Trans.) US: Cambridge University Press (Original 1st edition published in 1781, 2nd edition in 1787).
Katz, J. (1998). Realistic rationalism. Cambridge: The MIT Press.
Kitcher, P. (1984). The nature of mathematical knowledge. NY: Oxford University Press.
Kitcher, P. (1988). Mathematical naturalism. In W. Aspray and P. Kitcher, (Eds.), History and philosophy of modern mathematics. (pp. 293-325). Minneapolis: University of Minnesota Press.
Poincaré, H. (1952). Science and hypothesis. US: Dover Publications.
Putnam, H. (1975). Mathematics, matter and method: philosophical papers. (Vol. I). Cambridge: Cambridge University Press.
Quine, W. V. O. (1953). Two dogmas of empiricism. From a logical point of view. (pp. 20-46). Cambridge: Harvard University Press.
Rosado-Haddock, G. E. (2008). The young Carnap’s unknown master: Husserl’s influence on Der Raum and Der logische Aufbau der Welt. US: Ashgate.
Rosado-Haddock, G. E. (2006). Husserl’s philosophy of mathematics: its origin and relevance. Husserl Studies, 22, 193-222.
Verlade, V. (2000). On Husserl. US: Watsworth.
