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.
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.
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.
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.
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. 😀
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".
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 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.
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.
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.
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).
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"):
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):
"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.
"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.
"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.
"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.
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.
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.
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Por necesidad académica, y para proveer un buen resumen a mis estudiantes de Introducción a la Filosofía en Cayey, he preparado un material que resume algunas de las aportaciones más importantes y fundamentales del filósofo Gottlob Frege. Siguiendo mis propias convicciones en cuanto a material educativo, publico este material bajo tres licencias de documentación libre:
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