The Relation between Formal Science and Natural Science

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:

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This is how it looks like:

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You can look at the new website by going to 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, 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

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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Edmund Husserl

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) (ab) = 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:

  1. 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).
  2. 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. 😀


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.

A Journey to Platonism with Edmund Husserl — 11

On April 28, 2011, in Philosophy, by prosario2000

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.


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.

A Journey to Platonism with Edmund Husserl — 10

On April 23, 2011, in Philosophy, by prosario2000

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).

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A Journey to Platonism with Edmund Husserl — 9

On April 21, 2011, in Philosophy, by prosario2000

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
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

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.

Husserl's Theory of Logical Strata

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.


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.

A Journey to Platonism with Edmund Husserl — 7

On April 19, 2011, in Philosophy, by prosario2000

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:

  1. The morning star is a planet.
  2. 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.


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:

  1. Sensible Objects: pencils, houses, notebooks, computers, and so on.
  2. 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 :-P. 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.

Table of Husserl's Sense and Referent Doctrine

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.


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.

A Journey to Platonism with Edmund Husserl — 3

On April 11, 2011, in Philosophy, by prosario2000

A Letter to a Friend …

Carl Stumpf
Carl Stumpf (1848-1936)

One of the most intruiguing documents to date is a letter Husserl sent to his friend and mentor Carl Stumpf, which is key to understand Husserl’s mind from 1890 to 1891. Stumpf was a disciple of Hermann Lotze, just as Frege was, but he was also Brentano’s disciple, as Husserl was. Both, Husserl and Stumpf, were close friends, to the point that Husserl dedicated his philosophical masterpiece (Logical investigations) to no less than Stumpf. Here is the dedication.

with Honour and Friendship

Very simple words, yet they express everything.

Stumpf was very attracted to the philosophies of both Lotze and Brentano. Hermann Lotze was an intellectual child of German idealist philosophers (such as Johann Gotlieb Fichte, Friedrich Wilhelm Joseph Schelling, and Georg Wilhelm Friedrich Hegel). These philosophers were decisive in shaping his views about psychology and also an independent realm of logic and mathematics. Regarding the latter issue, though, he was more an intellectual child of G. W. Leibniz, a metaphysician of the seventeenth and eighteenth centuries. Leibniz was above all a great metaphysician and mathematician. He was Isaac Newton’s rival regarding science, and to a personal level too. Leibniz was one of the developers of calculus, which he elaborated independently of Newton … also a developer of calculus. When Leibniz published his work first, Newton then accused Leibniz of plagiarism, which led to a whole set of struggles between both after that. If you wish to know more about the Newton vs. Leibniz philosophical struggles, I suggest you read the chapter "The Work-Day God and the God of the Sabbath" in Alexandre Koyré’s From the Closed World to the Infinite Universe. Stephen Hawking’s A Brief History of Time also reports that when Leibniz died, Newton expressed his pleasure of finally breaking his heart. Umm…. Newton could be a jerk many times, despite his genius.

Anyway … Leibniz was a great mathematician. And one of the things he saw in a prophetic vision that geniuses have was that formal logic could be "mathematized". You can develop a sort of "calculus of propositions", just like mathematics. And, in fact, you could join mathematics and logic into an even bigger field he called mathesis universalis, the most universal mathematics of all.

This vision was crippled within the field of philosophy by Immanuel Kant’s Critique of Pure Reason. Kant was a better epistemologist than Leibniz, but unfortunately in the realm of logic and mathematics, he had no vision of the future. He said that formal logic belonged to the realm of analytic (a priori) judgments. On the other hand, in an unusual and weird way, he said that mathematical judgments belonged to the synthetic-a priori realm. Which means that logic and mathematics can’t mix in one sole place nor form together anything bigger. Many psychologists (i.e. those advocating for psychologism) did not accept Kant’s division. Many of them, including John Stuart Mill, conceived logic and mathematics as belonging to only one place: the synthetic-a posteriori realm, period. But some other not well known philosophers went the other way …

… Hermann Lotze was one of those philosophers (the other being Bernard Bolzano). Not only did Lotze believe in Leibniz’s dream of a mathesis universalis, but he also believed that mathematics could be derived from logic. This position is known as logicism.

Frege was Lotze’s disciple, and the latter was a very big influence on the former. One of the reasons Frege developed his conceptual notation (Begriffsschrift) was because he adopted a modality of logicism: arithmetic can be derived from logic, both arithmetic and logic are analytic-a priori disciplines. Frege’s conceptual notation is nothing more than an effort to mathematize logic to prove his logicist standpoint.

When Stumpf learned about Frege’s Begriffsschrift, he asked Frege to write a philosophical foundation for such symbolic notations (1882). Frege eventually did, when he published The Foundations of Arithmetic in 1884.

But Husserl was also a child of Leibniz, and although not a logicist himself, as a mathematician he saw a close association between logic and mathematics, to the point of positing the existence of an arithmetica universalis from a psychological standpoint. When he became a platonist, he was an advocate of a Leibnizian mathesis universalis.

Yet, Husserl expressed many of his colleagues, including Stumpf, that he was increasingly tormented about some stuff he can’t get out of his mind. And in a letter, he told Stumpf, just after Philosophy of Arithmetic went to press (1890), but before it was published (1891), that he could no longer hold as true the idea that numbers are reducible to sets, and that the whole of arithmetic could not reduced to cardinal numbers. I imagine that the first reaction by Stumpf went along the lines of … "What the heck?! Just when you are about to publish your book?!! Are you crazy?!!!" But Husserl was right!

"Imaginary" Numbers

Think about it! What are cardinal numbers? A quick refreshing course in number theory is in order:

Natural Numbers: 1, 2, 3, 4, 5, 6, … etc.

Cardinal Numbers: 0, 1, 2, 3, 4, 5, 6, … etc. (the difference between natural and cardinal is that the latter includes zero).

Negative Numbers: defined as a number that produces zero when it is added to the number. For example, you add -7 to 7, you’ll have zero as a result. So, -7 is a negative number. 7 + (-7) = 0

Integers: defined as the set of negative numbers and cardinal numbers. … -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7 …

Rational Numbers: Numbers which can be expressed through a ratio (a fraction or division). For example, the number two can be expressed as an interger (2) or as a ratio: 2/1. Also, a half would be a rational number (1/2).

Irrational Numbers: These are numbers which can never be expressed as ratio fraction. Usually, when expressed in decimals, irrational numbers tend to be a series of non-repeating patterns of numbers. Two of the most famous are the square root of two and the number pi.

Square Root of Two

Real Numbers: It is the set of rational numbers and irrational numbers.

Imaginary Numbers: An imaginary number is one number followed by a negative root, and by negative root, I mean the square root of -1:

Root of Negative One

For practical reasons, and to avoid confusion, if you have something like, let’s say the negative root of four, you analyze the root by expressing the negative four in terms of

-4 = (4) (-1)

Then you solve the square root of four, and represent the root of negative one as the letter i. So, the negative square root of four, is identical with 2i. This would be an imaginary number.

Yet in Husserl’s philosophy, he didn’t use many of these terms. When he used the term "real numbers", he was not referring to what we just described, but to all numbers which can be given to us on the basis of a set, a "collection" of objects. If this is the case, think about all of the numbers Husserl would have to leave out of mathematics:

  • Fractions
  • Negative Numbers
  • Irrational Numbers
  • Imaginary Numbers (from here on, I will refer these as "negative roots")

This would practically throw one very big chunk o mathematics out the window … it would be actually a very big step backwards to ancient times. Why?

You are never given fractions of anything. If you consider an object as a whole, let’s say a pizza, you can divide that pizza and form two halves of a pizza … right? (That’s the famous elementary school example when they teach you fractions). Yet, now that you have divided a pizza in two, what is going to prevent you from conceiving one half of that pizza as a new totality or whole?

Where do you see negative numbers if you base actually given numbers as a set? Have you seen negative sets? How in the heavens will you be given a negative set?

And let’s not talk about the utter impossibility of irrational numbers or negative roots to be given as sets!

This is the reason why, in his psychologistic phase, Husserl used a controversial term to refer to fractions, negative roots, negative numbers, irrational numbers, and the like. He thought of them as "imaginary numbers" (not in the same sense as negative root, but in the sense of numbers created in our own heads but not actually given in experience, as in the case of his version of real numbers).

Yet, there was something fishy about this conception of numbers. If all of these numbers are imaginary … why do they work? Why are they so wonderful to use for scientific purposes? What makes them tick? This was a big headache for Husserl.

You could say that negative numbers, fractions, etc. are a very clever use of numbers at the symbolic level, that they don’t refer to actual formal structures in objectualities at all, that they are just conceptual tools. Yet, Husserl was not a formalist. Numbers are not mere symbols or signs. Signs are the means to express them symbolically, but these signs have meaning. And if these numbers work, then that means that they must refer to some formal structures which are constantly applied to real objects in our mathematical calculi. THAT would explain why do they work (and work so well!)

In other words, the only way to explain why imaginary numbers work is by recognizing that they are not imaginary at all. At least from a mathematical standpoint, they are as real (in Husserl’s sense of the word "real" in this context) as real (cardinal) numbers are.

If this is true … then two conclusions are inevitable.

A Non-Reductionistic Approach

Husserl attempted to reduce all of arithmetic to cardinal numbers, and these to set theory. None of that worked. Yet, it cannot be said that sets, cardinal numbers, negative roots, fractions, irrational numbers, and so on are not completely valid, and cannot be used scientifically.

So, if these numbers work in every case they are applied, then two conclusions are the following:

  1. The only thing that can explain why numbers in general work so well in the world is that they must exist in some way, shape, or form. They cannot be merely the results of mental processes.
  2. No formal structure can be reduced to the other. Numbers cannot be reduced to sets, sets cannot be reduced to numbers. In fact, if you pay attention to what I said (regarding the pizza example) know that wholes cannot be reducible to sets or numbers, nor are the parts. Nor can ordinal numbers (first, second, third …) can be reducible to cardinal numbers. Irrational numbers, negative roots, fractions, and so on cannot be reduced to each other …

So, we are here at a very novel notion of mathematical objects. Many mathematicians and philosophers have tried (and many are still trying today) to reduce one kind of mathematical object to the other. Husserl rejected all of that. For him, these formal-objectual structures (or categorial forms, as he called them later), are all non-reducible to one another. These categorial forms include (but not limited to):

  • Sets
  • Ordinal Numbers
  • Cardinal Numbers
  • Irrational Numbers
  • Negative Roots
  • Parts and Whole

For Husserl, each of these is as important and fundamental as the rest.

By the way, this recognition of parts and whole as something non-reducible to set theory led to Husserl’s writing of the third investigation of Logical Investigations, whose reading led Stanislaw Lesniewski’s development of a new mathematical field: mereology (the science of the whole and parts).

In the Cambridge Companion to Husserl, there is an article dedicated to this neglected part of Logical Investigations, by Kit Fine. I highly recommend it. Guillermo Rosado Haddock, my friend and mentor, states in a review that one of the things Fine says in an endnote is that in the Third Investigation, Husserl seems "to foreshadow the structure of a relative closure algebra and, thus, that of a relative topological space" (Smith & Smith, 1995 , p. 475; Rosado, 1997, p. 385).

Imagine that!


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. US: Cambridge University Press. 

Frege, G. (1972, Jul.). Review of Dr. E. Husserl’s Philosophy of arithmetic. Mind, 81, 323, 321-337. (Originally published in 1894). 

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.

Husserl, E. (2003). Philosophy of arithmetic: psychological investigations with supplementary texts from 1887-1901 (D. Willard, Trans.) Dordrecht: Kluwer Academic Publishers. (Originally published in 1891).

Rosado Haddock, G. E. (1997, July-December). Edmund Husserl: a philosopher for all seasons? Modern Logic, 7, 380-395.

Smith, B. & Smith, D. W. (Eds.) (1995). The Cambridge Companion to Husserl. Cambridge: Cambridge University Press.

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A Journey to Platonism with Edmund Husserl — 2

On April 11, 2011, in Philosophy, by prosario2000

Important Note: I wish to say that I am not a Brentano specialist. I only make an exposition of part of his philosophy as I understand it, so, I will be very grateful for critical feedback on this content. Also, I’m not a specialist on the young Husserl, but I have read the Philosophy of Arithmetic, and some second sources, and expose it to the best of my ability, but at least I rely on recognized Husserlian scholars to interpret them.

Being Franz Brentano’s Disciple

Franz Brentano

Franz Brentano (1838-1917) was one of the most renowned philosophers of his day. Yet, he was not very appreciated in some circles because of one thing: he was a Jesuit priest. He had an overwhelming interest in Medieval scholastic philosophy to solve contemporary problems. Yet, as it turned out later, he was right to do so. Unfortunately, due to the modern prejudice that the Middle Ages are … umm… the "Middle Ages", and that it also is the "Dark Ages", then that means that there is no way you could build a modern philosophy on top of that! I mean, we know that Medieval philosophy was mixed with theology, and theologians like St. Thomas Aquinas wondered about how many angels can dance on the head of a pin?

Yet Brentano was interested in it, especially the way it dealt with Ancient philosophy, such as Aristotle’s. Now there is one heck of a prejudice of his contemporary philosophical colleages! They would have said somethin like: "A person interested in superstitious, dark, non-sensical Medieval philosophy to serve as lens to a more obsolete Ancient philosophy? Go figure!"

But at the same time, he was sympathetic with modern philosophy as well. He was a deep admirer of John Stuart Mill. This philosopher was the one who continued the tradition known as psychologism. Now, psychologism can mean many things. In my previous post I gave you one of them, but I’ll remind you:

  • Psychologism, in its original form (as formulated by John Locke), means that ideas (subjective content in your mind) are the referent of all knowledge. In a very real sense, you cannot know objects as they really are, only objects as they appear to you.
  • Regarding logic: psychologism is considering logic as being a branch of psychology. If logic is the science of correct reasoning, and psychology is a science of mental processes, ergo, logic must belong to psychology.
  • Regarding mathematics: all mathematical objects and operations are reducible to mental processes and representations.

Brentano was interested in this, but wanted to give the whole thing a new twist to it using a Medieval concept: intentionality. He expressed this philosophy in his Psychology from an Empirical Standpoint.

One of the big researches Brentano did at his time had to do with the concept of truth. He deeply admired Aristotle and his correspondentist conception of truth. Aristotle’s theory goes like this: If you say that x is a fact, when in reality it is not, or if you say that x is not a fact, when in reality it is, then you are making a false statement; but, if you say that x is a fact, and it is, or if you say that x is not a fact, and it is not, then you are telling the truth. Simple … right?!

Well … not that simple! Everyone at some level agrees with that, but it has problem… what does agree with reality? What are "facts"? What is "reality" anyway? Most people appeal to intuition when saying this, but when you dig up these concepts in philosophy, we find out that they are not always as precise concepts as they should be. Hence, the bad news is that as long as these concepts are imprecise, the concept of truth is just as imprecise or worse.

Along with questions regarding angels dancing on the head of a pin… Medieval philosophers and theologians alike were dealing with very serious questions about the nature of truth. These were the discussions Brentano was so worried about. And this was not easy at all for him. In his personal life, he gave up his priesthood after Council Vatican I declared the dogma of papal infallibility, which he argued, was inconsistent with Scripture and Tradition (something that I personally fully agree). This meant giving up his tenure in 1873. Much later he wanted to marry, a big problem at that time, since his Austrian citizenship forbid him from marrying anyone who were a former priest. He had to give up his citizenship and his professorship in 1880 so that he could marry. He was later allowed to teach as Privatdozent.

On the bright side, to further his research, he established what is known today as the School of Brentano, a set of students and philosophical researchers who wanted to explore his line of thinking. Belonging to this school were Alexius Meinong, Christian von Ehrenfels, Anton Marty, Carl Stumpf, Kasimir Twardowski, and … Edmund Husserl.


As I said, Brentano was looking for an adequate correspondentist view of truth, and he thought he could do that from a psychological standpoint (psychologism). For that, he wanted to use the concept of intentionality. What is intentionality? Very simple: it is that capacity of the mind to be directed to a particular objectuality. An objectuality is the referent of our intentional acts, something our mind refers to. Intentional acts include but are not limited to: desiring, loving, hating, thinking about, be concerned about, and so on. Each one of these mental phenomena are intentional: you desire something, you love something, you hate something, etc. That "something" being referred to by these mental acts is a particular objectuality, the referent of my desire, love, hate. An objectuality can be real or imagined. I can desire a crystal castle floating on the clouds, for instance … my desire is directed towards it.

So, what is truth? We have to specify what corresponds to what. If we are starting from a psychological standpoint (psychologism), this means that judgements arising from our of our mind are one of the poles of correspondence. Clearly, if I say that "I was born a Tuesday" or "I desire a crystal castle", these judgments must correspond to something else in order to be true. But what is the other pole of truth? Hard to say. Brentano changed his mind constantly regarding what the referent of our intentional acts is. Yet, one thing seems to be important: the concept of evidence. Evidence means that something is intuitively given to our self in an immediate manner. Psychologism regards this as a particular psychological trait, and one which should serve as one foundation of knowledge. So, the other side of the correspondence pole might be considered an idea as an objectuality, a referent of the judgment originated in our minds.

From a psychological standpoint we have to make a difference between the world as we perceive it (the psychological realm), and the world as it is (the physical realm). We only have access to the former, but not to the latter. Why? Because what we perceive is in reality the result of a whole set of processes within our body and our minds, hence not always our perceptions will correspond to what is "out there". We only have full access to our inner "life" so to speak, our presentations [Vorstellungen] in our minds. Yet, when we say a judgment like "I was born a Tuesday", we are actually stating a fact … we are saying that this is the case. In this sense, all judgments are existential ("this exists" or "this does not exist"). Yet, the other pole of the theory of judgment seems to be presentations, mental representations in our minds, and that’s it! So, judgments arising in our heads correspond to ideas also arising in our heads. What guarantees the objectivity of that?! This represented a whole set of problems for Brentano, which is the reason why he established a school about that issue.

One of his problems, of course, is that he recognized the objectivity of mathematics, logic, and truth … but he didn’t know exactly how to make all of these fit in his philosophy.

… That’s where Edmund Husserl came in … as I said in my previous blog post … he had the training …. and what a mathematical training that was! A disciple of Kronecker, Königsburger, and Weierstrass? Are you kidding! Brentano loved Husserl’s presence in his school! Also he was excited at the fact that Husserl was also interested in the concept of number from a psychological standpoint. His inspiration was Brentano himself. He was very kin to Brentano’s notion of intentionality, and wanted to give it a try.

On the other hand, there was a new mathematical theory being developed at the time: set theory, initiated by no less than Georg Cantor, who was already Husserl’s close friend and mentor at the time. Husserl called sets "collections", "totalities", or "multiplicities". Essentially you can imagine any group of objects as sets: a set of mountains, or a set of students, or a set of pencils.

Also, from an intuitive standpoint Husserl thought that sets (or collections or multiplicities) were the basis of the concept of number, which we can represent in a variety of ways. The number one, for instance, can be respresented by the word "one", or the Indo-Arabic numeral "1", or the Roman numeral "I". Each one of these signs refer to a set of one object. All of the signs for two would represent a set of two objects, and the same is true of other signs.

On the other hand, we never see sets themselves, nor numbers themselves, just the objects we perceive with our eyes or can touch with our fingers. This means that sets and numbers are of psychological origin, a result of abstracting from experience in some way. Husserl began this enterprise on this premise, using his teacher’s notion of intentionality.

How Do Numbers Come to Be? (Not Really a Solved Matter, Though… )

In his philosophical enterprise, Husserl wanted to address three questions:

  1. What is the number itself?
  2. In what kind of cognitive act is number itself actually present in our minds?
  3. How do the symbols or symbolic systems used in arithmetical thought enable us to present, and to arrive at knowledge of, numbers and numbers relation that are not … intuitively given to minds such as ours … and even enable us to have the most secure knowledge possible concerning many of the properties of and relationships between the larger numbers? (Husserl, 1891/2003, p. xiv)

For Husserl, the collections of objects are the basis for cardinal numbers. As you know, cardinal numbers are the natural numbers (1, 2, 3, 4, 5, … etc.) including zero. So, an empty group equals zero, a collection with one element is one, a group of two objects is two, a group of three objects is three, and so on. "Clearly" the notion of number arises from mental acts (as we have seen before), but taking sensible objects (physical or imaginary) as a basis for sets, and at the same time, these sets are the basis of cardinal numbers. We have to consider each set or collection of objects as a totality, as a "whole", of which subcollections of elements or the elements themselves are considered "parts".

For him, groups of objects are given intuitively as unified wholes as a result of a mental act, or a reflection on the objects being presented to us in our experience. Also we must point out that for Husserl, also numbers present themselves in our experience. I think Dallas Willard explains this very well:

If … I attempt to count the trees in a certain area of the park, I … must do something more than just be conscious of them, or even clearly see them… I must rather, as I view them, think the characteristic manner: There is that one and that one and that one and … . As I go through these acts in which things are enumerated are "separately and specifically noticed," as Husserl says, there arises for me a division of the trees into those "already" enumerated and those not or "not yet" enumerated. [Husserl’s] view is that this division is an objective fact intuitively given to me. If it does not present itself to me with some force and clarity [i.e. evidence], I simply cannot number the trees. But in that it does come before me as I count, the trees already enumerated appear "together," and in their unification with each other they stand "apart from" the remaining trees and objects — of which I nonetheless may be quite conscious all along. The "number of things," the "totality" or "multiplicity" — a different one at each step as I count — is intuitively "constituted" (made present) for me in this type of thoughtful enumeration. (Husserl, 1891/2003, p. xix-xx).

Yet, the way this collection (or set) appears is through a collecting act of the mind, which Husserl termed a "psychical" or "psychological" relation. Each element is conceived as "something" (it seems to be his philosophical version of a mathematical variable), which is the result of an abstracting process from sensible objects, a result of what Husserl termed "reflexion". Husserl describes it this way:

We obtain the abstract multiplicity form belonging to a group by diminishing each of its elements to a mere "one" [or "something"] and collectively grasping together the units thus originating. And we obtain the corresponding number by classifying the multiplicity form thus constructed as a two, a three, etc." (Husserl 1891/2003, p. 109)

Following Brentano, he thought that numbers are "collected" into a group by the concept of conjunction (the word "and" in every day sense: "this and this"). The number is the concept which designates any set of objects with similar formal structures. For instance, the number two designates the formal structures of different sets of objects "two apples", "two tables", "two chairs", etc.

But about the word "something", what does it mean exactly? For Husserl, this "something" is the result of reflexion on sensory objects either in the physical world or in the imagination.

The unification [of a multiplicity] comes about … only in the psychical act of interest and perception which picks out and combines the particular contents and can also be perceived in reflexion upon the act (Husserl, 1891/2003, p. 164).

Reflexion lets us conceptuate from experience. Yet, the concept of "reflexion" is not too clear in Husserl. He didn’t make a distinction between the sort of objectuality that lets us see the objects and the formal structures (sets) on the one hand, and on the other, the whole process of concept formation based on those objects along their formal structures. As we shall see, this crucial distinction is one of the major reasons that Husserl will renounce psychologism altogether.

Apparently also Husserl didn’t establish another clear distinction between a mental act of constituting these formal structures based on sensible objects and the objectuality it is constituting. Again, this will prove crucial to Husserl’s change of mind.

So … Was Frege Correct?

As we have seen in my previous post, Frege accused Husserl of turning everything in a presentation (Vorstellung), into subjective mental activity. Is this true? Well, there is a grain of truth to it in the sense that Husserl was constantly talking about "presentations". Yet, here is the problem: Frege used the word Vorstellung in a very special way. As Claire Ortiz Hill and other scholars like Rudolf Bernet, Iso Kern and Eduard Marbach have pointed out, for Frege, the term Vorstellung is used to mean subjective mental ideas which cannot be shared with other minds. My current representation of the unicorn in my mind is a Vorstellung in Frege’s sense, only I can have plenty of access to it. You can’t.

Is this what Husserl meant with Vorstellung? Not really. Husserl used the term presentation in a more conventional way, which could include an objective conception of "presentation". Hill shows these passages as examples:

If a totality of objects, A, B, C, D, is our presentation [Vorstellung], then, in light of the sequential process through which the total representation originates, perhaps finally only D will be given as sense presentation [Vorstellung], the remaining contents being then given merely as phantasy [i.e. imaginary] presentations [Vorstellungen] which are modified temporally and also in other aspects of their content. If, conversely, we pass from D to A, then the phenomenon is obviously a different one. But the logical signification sets all such distinctions aside. … In forming the presentation [Vorstellung] of the totality [i.e. the set] we do not attend to the fact that changes in the contents occur as the colligation progresses. Our aim is to actually maintain them in our grasp and to unite them. Consequently the logical content of that presentation [Vorstellung] is not, perhaps, D, just-passed C, earlier-passed B, up to A, which is the most strongly modified. Rather, it is nothing other than {A, B, C, D}. The presentation [Vorstellung] takes in every single one of the contents without regard to the temporal differences and the temporal order grounded in those differences. (Husserl, 1891/2003, pp. 32-33).

Now, it seems that Husserl is making here two different sorts of presentations or Vorstellungen: one subjective, the imaginary presentations (such as the ones Frege is talking about), and another the actual presentation, which is the sensible object itself "in person", so to speak. An even clearer example that not everything is reduced to subjective mental activity can be seen here:

Certainly one distinguishes in complete generality the relating mental activity from the relation itself (the comparing from the similarity, etc.). but where one speaks of such a type of relating activity, one thereby undderstands either the grasping of the relational content or the the interest that picks out the terms of the relational content or the interest that picks out the terms of the relation and embraces them, which which is indispensable precondition for the relations combining those contents becoming observable. But whatever is the case, one will never be able to maintain tha the respective act creatively produces its content. (Husserl, 1891/2003, p. 44).

In other words, it doesn’t matter which imaginary or mental activity we engage in, our minds will not create or modify the actual objects being shown to us "in person". So much for Frege’s "cats which become phantoms" portrayal of Husserl’s psychologistic theory. So much for years of analytic and continental philosophers portraying Frege’s view of Husserl’s psychologism as "biting and accurate" (e.g. Coffa, 1991, pp. 68-69).

Despite the way Frege tried to present it, Husserl’s psychologism is in fact a moderate version, much more moderate than Brentano’s. Why didn’t he fall into Brentano’s version of psychologism? Very simple, and so extremely obvious at this stage that people completely ignore it: Husserl was a mathematician!!!!!

If anything, he aspired to rigor of thought, as every mathematician does! He sympathized a lot with his teacher, Brentano, and loved him dearly. He wanted to follow his footsteps … but then came Husserl the mathematician, who tried to establish a psychological origin of number from a philosophical perspective, while, at the same time, asserting that numbers are objectually given at once along sensible objects in any given mental act. It is as if there were a distinction between the subjective activity of the mind, and the number itself! As if the number is not actually originated in our minds, only given! And there … as they say … lies the rub!

In one aspect Frege was correct, even when incredibly careless, that Husserl was using the psychological term Vorstellung. Husserl came to recognize that he was using the term equivocally, and he even regretted his psychological approach, which he considered "immature".

So, whatever Frege did to change his mind, it seems as if he didn’t change him much. You only change your views if they are portrayed correctly, and are adequately refuted. Frege was correct in some aspects of Husserl’s philosophy in his (in)famous review, but not in the crucial passages we have shown before.

And, as it turns out, if Frege influenced Husserl at all, it was with The Foundations of Arithmetic. But this philosophical gem was not enough to change his mind. Later, in his Introduction to the Logical Investigations, an a posthumously published draft, Husserl would name the philosophers who were a decisive influence in making him change from psychologism to platonism: G. W. Leibniz, Bernard Bolzano, Hermann Lotze, and David Hume. Frege is not even mentioned at all. He read these philosophers in the year 1890 … long before 1894, the year of Frege’s review. And by the time his Philosophy of Arithmetic was available (1891), Husserl had already changed his mind … And guess what … as early as 1891, Frege implicitly already identified Husserl as a platonist.


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. US: Cambridge University Press.

Frege, G. (1972, Jul.). Review of Dr. E. Husserl’s Philosophy of arithmetic. Mind, 81, 323, 321-337. (Originally published in 1894).

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.

Husserl, E. (1975). Introduction to the Logical investigations: a draft of a preface to the Logical investigations. The Hague: Martinus Nijhoff.

Husserl, E. (2003). Philosophy of arithmetic: psychological investigations with supplementary texts from 1887-1901 (D. Willard, Trans.) Dordrecht: Kluwer Academic Publishers. (Originally published in 1891).

Künne, W. (2003). Conceptions of truth. Oxford: Clarendon Press.

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A Journey to Platonism with Edmund Husserl — 1

On April 5, 2011, in Philosophy, by prosario2000

A Journey to Platonism with Edmund Husserl — 1

Edmund Husserl

A Terrifying Frustration and a Definitive Turn

It was a terrifying evening in 1894. Husserl was listening to the wind howling outside, and the thunder clapping in the sky. The bats were flying all over the house, and the full moon was outside. He was traumatized, crying tears out of despair. He was reading a review on his most recent book The Philosophy of Arithmetic, and he saw all of his philosophy collapse in his mind. He looked at criticisms like this one:

… In the case of the word "number", for example, the aim is to exhibit the appropriate presentation and to describe its genesis and composition. Objects are presentations [Vorstellungen]. … Since everything is now a presentation, we can easily change the objects by now paying attention, now not. We pay less attention to a property and it disappears. … For example, let us suppose that in front of us there are sitting side by side a black and a white cat. We disregard their colour: they become colourless but are still sitting side by side. We disregard their posture: they are no longer sitting, without, however, having assumed a different posture; but each one is still at its place. We disregard their location: they are without location, but still remain quite distinct. Thus from each one we have perhaps derived a general concept of a cat. Continued application of this process turns each object [into a more and more bloodless phantom]. …

… the difference between presentation and concept, between presenting and thinking, is blurred. Everything is shunted off into the subjective. But it is precisely because the boundary between the subjective and the objective is blurred, that conversely the subjective also acquires the appearance of objective. …

… In combining under the word "presentation" [Vorstellung] both what is subjective and what is objective, one blurs the boundary between the two in such a way that now a presentation in the proper sense of the word is treated like something objective, and now something objective is treated like a presentation. Thus in the case of our author, totality (set, multiplicity) appears now as a presentation, now as something objective. …

… According to the author [Husserl] a number consists of units. He understands by "unit" a "member of a concrete multiplicity insofar as number-abstraction is applied to the latter" or "a counted object as such". … In the beginning, the objects are evidently distinct; then, by means of abstraction, they become absolutely the same with respect to one another, but for all that, this absolute sameness is supposed to obtain only insofar as they are contents. … (Frege, 1894/1972, pp. 324-325, 331)

And Husserl cried out to the heavens: "Why have I been such a moron?! Why?! Whyyyy?!!!!"

OK! ……OK! …… OK! … I grossly over-dramatized the whole thing! Yet, many people around the world, including many Husserlian scholars believe in something similar (in a much less dramatic sense). Needless to say that many analytic philosophers, who praise Gottlob Frege over Edmund Husserl also believe this. And who is Gottlob Frege?

Gottlob Frege

Frege (1848-1925) happens to be a major figure in contemporary philosophy, because he was one of the most eminent philosophers who developed symbolic logic or mathematical logic. He wrote a small book on that subject called Begriffsschrift (Conceptual Notation), where he proposed a conceptual notation or a symbolic proposal to establish relationships among propositions, which could serve as basis to prove that arithmetic can be derived from formal logic. Later, one philosopher who would become Husserl’s friend and mentor, Carl Stumpf, asked Frege if he could elaborate the philosophical basis for that notation, and he agreed. Frege wrote The Foundations of Arithmetic, which should be considered one of the finest gems of analytic philosophy. I highly recommend its reading!

In it, he engaged against forms of psychologism, and naturalism. In this context, "Psychologism" is the philosophical view that the validity of mathematical truths must rely in psychological processes of sensory abstraction, and "naturalism" means that mathematical concepts such as numbers, ultimately refer to natural or physical objects. There are other trends, such as empiricism, in which mathematical concepts tell us about aspects of experience; constructivism, in which numbers are constructed by our minds; anthropologism, in which mathematical objects are socially constructed within a culture; and formalism where mathematical symbols are treated as signs and nothing more than that.

Frege proposed a realist view of mathematics: realism means that mathematical concepts refer to existing abstract mathematical objects. The sort of realism proposed by him is called platonism: which means that these abstract mathematical objects exist independently from the physical and psychological worlds.

Note: Platonism in this context does not mean, Ancient Platonism, where abstract ideas are the archetypal forms from which the physical world participates. Contemporary platonism just means that mathematical objects such as numbers, sets, and others, have an independence from the physical or psychological worlds. That’s it.

Husserl’s Background

By the time Frege wrote The Foundations of Arithmetic (1884), Husserl was a partisan of psychologism, which is the reason why Frege went after him in his review. Husserl had written his Habilitationsschrift titled On the Concept of Number (Über den Begriff der Zahl) in 1887, under Stumpf’s supervision while he was in the University of Halle. Later, this served as a basis for Husserl’s work Philosophy of Arithmetic. Both works espoused a form of psychologism. For Husserl, numbers have no independent existence from the mind.

Despite the differences of opinion with Frege, Husserl sent him a copy of Philosophy of Arithmetic, and a review on a book by E. Schröder. The reason for this is that Husserl spent some pages of his latest work commenting on Frege’s The Foundations of Arithmetic. This criticism did not translate in absolute hostility towards Frege, quite the opposite. Along with the copy of his book, he sent a letter where Husserl told him that no other book had provided him with nearly as much enjoyment as The Foundations of Arithmetic. He was stimulated by Frege’s work, and according to Husserl, he derived constant pleasure from the originality of mind, clarity and honesty. According to Claire Ortiz Hill, in the Philosophy of Arithmetic, Husserl cites Frege more than any other author mentioned in his work. And in a letter, Frege recognized that Husserl’s study of his Foundations was perhaps the most thorough one that had been up to that time (Hill & Rosado, 2003, p. 4).

Yet, we need to clarify something here, because Frege’s treatment of Husserl as a moron is an enigma in light of all of this. First, Husserl was not a moron at all. Both he and Frege were professional mathematicians. This has been emphasized in the case of Frege, but not enough in Husserl’s case. Let me remind you about it.

  • While Husserl studied in the University of Vienna, he studied mathematics under the supervision of Leo Königsberger (1881). Königsberger (1837-1921) was a former student of Karl Weierstrass, and who made many contributions to the research of integral calculus and differential equations.
  • While Husserl was a student at the University of Berlin, he studied with mathematicians such as Leopold Kronecker and Karl Weierstrass (1878-1881). Later, Husserl became Weierstrass’ assistant (1883-1884). Kronecker (1823-1891) contributed to theory of equations, and the concept of continuity. He also solved the quintic equation applying group theory. Weierstrass (1815-1897) was more eminent. Calculus students owe him a lot, because he was the one who formalized the concept of uniform limit and uniform convergence in functions, and applied them to the whole of calculus. This also set the basis for further mathematical discoveries of his own.
  • It was in the University of Halle where he befriended Georg Cantor, who would become his mentor (1886-1901). Who is Cantor (1845-1918)? Ah .. just the father of set theory, which is integrated at the basis of contemporary mathematics.
  • He was also a friend of Ernst Zermelo (1871-1953), whose major works include research on set theory. During that research, in 1902, he discovered the so-called "Russell Paradox" before Bertrand Russell. When this happened, he sent a letter to Husserl talking about it, because in one of his reviews, Husserl talks about a similar paradox (not the same one, though).
  • Later, when Husserl went to the University of Götingen, he was a close colleague of David Hilbert, and formed part of Hilbert’s Circle (1901-1916). David Hilbert (1862-1943) made great advances in mathematics, especially regarding the axiomatization of geometry and functional analysis.
  • Husserl studied Non-Euclidean geometry extensively, and was acquainted with the works of Bernard Riemann, and his notion of manifolds. He also made serious research on set theory, and his phenomenological doctrine seems to establish a sketchy and basic way to solve in principle two major paradoxes of set theory: Cantor’s Paradox, and the so-called Russell’s Paradox (it should be called the "Zermelo-Russell Paradox"). However, his mathematical and philosophical research on the paradoxes of set theory are still unpublished. :-S
  • Although not his subject, he also contributed to today’s distinction between "formation rules" and "transformation rules" in formal logic. These distinctions are attributed to Rudolf Carnap, but Husserl made it first. In fact, some people suspect that Carnap borrowed it from Husserl and changed the names of these rules.

Hmm… it seems that with this background Husserl was not such a moron after all. There seems to be a problem with the "Frege turned Husserl into a platonist overnight in 1894" scenario. After all, Husserl the mathematician seemed to spend his life on the side of the crême-de-la-crême in mathematics discoveries and research. He even met and befriended far more mathematicians than Frege and Russell.

Why did Husserl change his mind to platonism? How did he change his mind? What sort of platonism did he propose? That will be the subject of the following posts on this subject.


Frege, G. (1972, Jul.). Review of Dr. E. Husserl’s Philosophy of Arithmetic. Mind, 81, 323, 321-337. (Originally published in 1894).

Hill, C. O. & Rosado-Haddock, G. E. (2003). Husserl or Frege? Meaning, objectivity, and mathematics. US: Open Court.

Husserl, E. (2003). Philosophy of arithmetic: psychological investigations with supplementary texts from 1887-1901 (D. Willard, Trans.) Dordrecht: Kluwer Academic Publishers. (Originally published in 1891).

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Material de Frege Corregido (v.0.3)

On November 17, 2010, in Philosophy, by prosario2000

Gottlob Frege

Hace algunos meses atrás hice disponible un material sobre Gottlob Frege para propósitos educativos. Hago aquí disponible la versión 0.3 de dicho material. Hice correcciones de presentación de contenido. Como siempre, agradeceré a cualquiera que coopere en el proceso de corrección del material (escríbanme a Espero que sea útil.

Como es un libro de texto, considero esta obra como funcional, y la hago disponible bajo tres licencias copyleft: la licencia Creative Commons Attribution-ShareAlike 3.0 Unported, la GNU Free Documentation License y la GNU General Public License. Las tres licencias caen bajo las definiciones de Obra Cultural Libre y de Conocimiento Abierto. Por ahora, hago disponible el texto en PDF y en ODF (éste último para propósitos de modificación, debe considerarse código fuente). Se recomienda que se baje e instale las letras Linux Libertine para ver el documento correctamente en ODF.

Versión PDF del Texto

Versión ODF del Texto

Espero que el texto sea de su agrado.

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