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.

Material Educativo de Gottlob Frege (v. 0.35)

On April 24, 2011, in Philosophy, by prosario2000

Gottlob Frege

Decidí recientemente hacer una nueva edición de este material sobre Gottlob Frege, debido a que necesitaba sacar una versión nueva de Frege para mis estudiantes de Introducción a Filosofía. No es una modificación grande, solamente incluye corrección de expresión y de estilo, de tal manera que sea más fácil para los estudiantes comprender el material. Falta material que se elaborará posteriormente, pero para mis fines personales es suficiente.

Ya que tengo disponible una página de internet, este material de Frege va a tener su sede permanente en mi página, y pueden conseguir el material, tanto en formato fuente (ODF) como en PDF y DjVu. Aquí pueden acceder a la página.

Página de Internet
Página de Internet

Cualquier duda en cuanto a estos formatos (ODF, PDF, DjVu), consulte a esta página.

Como siempre hago disponible este texto bajo tres licencias copyleft:

Las tres licencias caen bajo las definiciones de Obra Cultural Libre y de Conocimiento Abierto. Para ver y usar el documento en ODF, se recomienda que se baje y se instale las letras Linux Libertine.

Cualquier crítica o sugerencia, por favor, escríbanme a

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A Journey to Platonism with Edmund Husserl — 8

On April 20, 2011, in Philosophy, by prosario2000

Problems with Sets

When set theory was first formulated by Georg Cantor and elaborated later by Ernst Zermelo, they noticed that the way the whole system was formulated gave way to paradoxes. Usually a paradox comes up when the system deductively lets certain cases derive contradictory propositions. One famous paradox (although not a set theory paradox) is called "The Liar Paradox": when a liar says "I’m always lying", is he lying or not? If he is lying then that means that he tells the truth (not always lying), but if he is telling the truth, then he is lying. It is a paradox so common since antiquity, that this appears in the Bible (Titus 1:12).

Set theory had two major paradoxes at the time.

1. Cantor’s Paradox

Georg Cantor and other mathematicians developed the concept of power set. It is the set of all the subsets of a given set. …. Ok! Ok! … I know, it sounds confusing, but let me give you an example. Let’s say that we have this set, let’s call it S:

S = {x, y, z}

Then the power-set (P) of S is the following:

P(S) = { {}, {x}, {y}, {z}, {x,y}, {x,z}, {y,z}, {x,y,z} }

In other words, we can take set S, and the power-set will tell you all of its possible subsets (lower order sets). The first one will always be the "empty set" (that is, a set with no elements), then each element of the set can serve also a subset. Then every possible pair of elements is a subset. Finally, all three elements themselves can be also a subset. Simple, right?! The power set is usually bigger than the original for this reason.

Now, here is the problem with what has come to be known as naïve set theory, with the Cantor Paradox:

Naïve set theory states that it is possible to form the set of all sets. If it is the "set of all sets", then there should not be any bigger set than that … right? But what about the power set? The power set of that set of all sets is by definition greater than the set of all sets … which would mean that the set of "all" sets is not really the set of all sets anymore :-P. Hence a paradox occurs.

2. The Zermelo-Russell Paradox

Frege published the first volume of its Basic Laws of Arithmetic in 1893 as the beginning of his proof that arithmetic is derivable from logic. Just when he was about to publish the second volume in 1903, he received a letter from Bertrand Russell telling him that his work was wonderful, but it had a problem: it allowed for a paradox, which is today known as the Russell Paradox. Today we know that Ernst Zermelo discovered it first and independently from Russell, hence I call it "the Zermelo-Russell Paradox". This paradox blew up Frege’s logicist enterprise to oblivion. I wouldn’t have liked to be in Frege’s shoes. Imagine that you carry out an entire life trying to prove something, and then find out that the enterprise itself was in vain in the end, despite Frege’s own contributions to logic, mathematics and semantics.

But what is this paradox about?

Let’s point out the fact that there are sets which are not part of themselves, or that they are not elements of themselves. What do I mean by that?

Let’s imagine a set of all cats. This necessarily goes from the black cat that you are so afraid to find on the street, to the Cheshire cat who tormented Alice. Imagine all cats grouped together in a set. That set itself is not a cat … right? Therefore, the set of all cats it is not part of itself.

Now, let’s imagine a set of all tables; this set does not form part of itself either … because the set of tables is not itself a table.

And we could go on, the set of all chairs, all TVs, all Presidents of the U. S. … you name it!

Now take all of these sets, and form the set of all sets which do not form part of themselves. This means, take the set of all cats, the set of all tables, the set of all TVs, etc. .. and form this huge "mega-set" of all the sets which do not form part of themselves.

This might seem plausible, right? Now, here is the paradox: Does this "mega-set" form part of itself or not? If it does not form part of itself, then by definition it should form part of itself. But if it does form part of itself, then, by definition, it does not!

After reading about these two paradoxes, I imagine you saying something like: "Wow! Apparently mathematicians have nothing else to do with their time!" Yet, these are not minor problems. … Ask Frege! He’ll tell you all about it.

Epistemology of Mathematics

As I said in my previous blogs, Husserl was not only concerned about logic or mathematics, but he was also concerned about knowledge.

His answer to the problem of knowing mathematical objects has a lot to do with the reasons why he left psychologism … and submitted all sorts of criticisms to empiricism and naturalism in general (which we’ll see in future blog posts). For now, suffice to say, that part of the reason why he left behind all sorts of naturalistic accounts for science is that the theory of knowledge was inadequate. In Ideas pertaining to Pure Phenomenology and to a Phenomenological Philosophy (1913) he criticizes psychologists (proponents of psychologism), naturalists and empiricists.

First of all, Husserl recognizes the enlightened spirit of philosophical naturalists in general, especially when they wish to eliminate all sorts of mysticisms and superstitions from philosophy. He says: "Hey, I get it! And guess what? … I totally agree". But in so doing, they do too much trying to extirpate all sorts of necessary aspects to all knowledge: such as, for instance, essences.

Naturalists are essentially anti-essences. They wish to extirpate essences because they form part of a Platonic heritage, which appeared also as part of Aristotle’s metaphysics. Aristotle apparently sunk the Middle Ages into … the Middle Ages :-P, or in pure darkness of ignorance … only to be rescued by rational and, especially, empirical and naturalistic thinking. Great!

Yet, Husserl points out that if you take a very good look to scientific theories and the way they have succeeded, they all rest in essences. What is an essence anyway? An essence is what is conceptually or logically necessary and always true or always false, no matter what. Logical and mathematical schemes are a network of essential relations among propositions made in scientific theories, which refer to a network of objects (in this case, observable phenomena). In the case of usual geometry, where we use points, lines, space, planes, and so on, we establish also material (not only formal) relationships among these concepts, and science also depends on this.

As the Emperor of Star Wars, we could say about essences: "There is no escape!"

Also, as we have stated in our earlier post, observable objects are not the only things given to us, but also their formal relations, the famous formal-objectual categories.

Therefore, the problem with naturalist epistemology is that it is incomplete, and it has to account for our knowledge of essences and formal categories.

Mixed-Categorial Acts

Husserl bases his theory of knowledge on intentional acts. Remember that intentionality is an act of our own consciousness which directs itself to an object.

Yet, as we have stated before, what is given to our consciousness is always a state-of-affairs. They are objectualities; they are referents of our intentional acts. Yet, as we have seen in our previous blog post, states-of-affairs have two very important components: material content (sensible content), and formal arrangement (formal-objectual categories). How are these constituted?

Husserl’s theory of knowledge is based on what he calls intuitions. An intuition is that aspect of our consciousness which gives us whatever our intentional acts are directed to. Let me explain this in English. For example, if I’m looking at the screen, I have an intuition of the screen being in front of me: it is given to me at once, it is there, and my thinking is directed to it (that is my intentional act).

For Husserl there are two sorts of intuitions involved in the constitution of an objectuality or state-of-affairs:

  1. Sensible Intuition: This is what makes possible the constitution of objects we see, hear, taste, smell and touch. He further identifies two sorts of sensible intuition: sensible perception, which involves the intuition of objects being given to us "in person" (so-to-speak), like this computer screen in front of me; and sensible imagination (my constitution of objects in my imagination: fairies, imaginary tables, crystal castles, and so on).
  2. Categorial Intuition: This is what makes possible the constitution of formal-objectual categories: numbers, sets, part-whole, relations, and so on. For him, there is also a distinction between categorial perception (when categorial forms are founded on objects "in person") and categorial imagination (when they are founded on imaginary sensible objects).

Each objectuality is the result of what Husserl calls an objectual act, the mental act of turning a state-of-affairs into our object of our consciousness. Each objectual act our consciousness carries out in the world or in our imagination is what he called: mixed-categorial acts. What does that mean? It means that these acts constitute a state-of-affairs in which we are given sensible objects along with their formal-objectual categories. I perceive three pencils, or a set of books, or Megan being taller than Mary, and so on. All of these constitutions of sensible plus formal states-of-affairs are the result of mixed-categorial acts.

Categorial Abstraction

Yet, when we deal with mathematics … pure mathematics … we do not deal with three pencils or a set of books. We deal with the number three itself, or the set. We don’t have to deal with: "two books and two books are four books in total" … but with "2 + 2 = 4". In fact, sometimes we deal with equations such as "x + 1 = 3" or even "x + y = y + x" or {x, y, z} and so on.

Here, Husserl introduces the concept of categorial abstraction, where intentionally we disregard all sensible objects given in sensible intuition, and just deal with the categorial forms themselves. This is what Husserl calls a pure-categorial act, with which we constitute pure categorial forms, they become our object of our consciousness. This is also called formalization.

So, when we talk of mathematical intuition in Husserl’s case, we are talking about categorial intuition plus categorial abstraction. That’s how we know abstract objects such as numbers, sets, and so on.

Eidetic or Essential Intuition

Yet there is a third form of intuition which Husserl calls eidetic or essential intuition: an intuition of essences. What does this intuition do? It gives us essential relations at once, we are able to recognize through understanding the universality and necessity of what is being given. Let me give you a very obvious instance of how this happens:

"x + y = y + x"

This equation is constituted by a pure-categorial act, the x and y are indeterminates (variables) which can represent any number whatsoever. Yet, when we see it, we instantly and immediately recognize that it is universally and necessarily valid. We don’t have to go through a tedious process of substituting x and y with different numbers to check if there is some arithmetical exception to this rule. No! Not at all! We know that there are no exceptions to this mathematical truth.

The same happens with some material concepts such as the essential relationships between lines, points, spaces we can see in traditional euclidean geometry. The same we can see in part-whole relations too.

Hierarchy of Objectualities

Now that we understand the three intuitions and mathematical knowledge, we are able to understand something regarding objectualities or states-of-affairs: that there can be hierarchies of objectualities or states-of-affairs.

Let’s say that I have these sets of books:

  • Set of Books on Quantum Physics
  • Set of Books on Cosmology
  • Set of Books on Zoology
  • Set of Books on Cells
  • Set of Books on Plants

Each of these sets is given to me thanks to the fact that there are books on each of these subjects. These books are the sensible objects, and I am able to constitute them as sets of books by means of objectual acts.

Yet through other objectual acts, I can still constitute a higher-level of sets! For example:

  • Set of Books on Physics (Sets of Quantum Physics and of Cosmology Books)
  • Set of Books on Biology (Sets of Zoology, Cells, and Plants Books)

Through another objectual act, I can even constitute a higher objectualities:

  • Set of Science Books

And I could continue indefinitely! If I wish to find out the sensible objects which are the reference basis for this hierarchy of objectualities or states-of-affairs, all I have to do is to trace down the different objectual levels to the sensible components.

If you want a hierarchy of pure objectualities, just get rid of the sensible components, substituting them by indeterminates, and you have a hierarchy of sets in its purity.

Husserl used sets to illustrate this because it is the easiest example, but this hierarchy is not limited to sets, it also extend to all other formal-objectual categories.

For more technical details on Husserl’s epistemology of mathematics, see Hill & Rosado (2000, pp. 221-239). The original information appears in Logical Investigations, second volume, Sixth Investigation, §§ 40-52, 59-66. In Experience and Judgment he also talks about the hierarchy of objectualities using sets as examples.

How to Solve Some Paradoxes

Do you remember the paradoxes of naïve set theory? Well … Husserl solved them in his philosophy.

Solution to Cantor’s Paradox: In principle, every set can be an element of a higher-order set. This itself blocks the possibility of a "set of all sets". It simply can’t happen in this mathematical scenario.

Solution to the Zermelo-Russell Paradox: The hierarchy of objectualities or states-of-affairs blocks the possibility of any set forming part of itself. In this scenario, no set can form part of itself.

If you wish more technical details on this subject see Hill & Rosado (2000, pp. 235-236).

The First Platonist Epistemology

For all practical purposes, Husserl proposed the first platonist epistemology of mathematics. Many others have really tried to do something similar and failed miserably. Frege, for instance, talked very loosely about "grasping" mathematical objects. Yet, his semantic doctrine reduced numbers to his notion of objects (saturated entities). Husserl conceived numbers as formal structures, making it possible to develop an epistemology of mathematics. Some other philosophers, like James Robert Brown, have talked about an exotic faculty of knowing numbers they call the "mind’s eye", but when examined, it really does not explain anything.

I wish to say, though, that Jerrold Katz tried to develop a similar epistemology in his book Realistic Rationalism, which is (in my judgment) a philosophical gem. For him, mathematical objects are also structures given along sensible objects, and our mathematical intuition consists of getting rid of those sensible objects. However, the difference between Husserl and Katz is two-fold. For Katz, mathematical intuition includes what Husserl would call eidetic intuition. Also, Katz’s theory, unfortunately, is not as clear and sophisticated as Husserl’s.

Finally, I want to point out that Husserl’s criticism to psychologism, empiricism, and naturalism, led him to enrich the current understanding of intuition. These three doctrines tended to limit our knowledge and intuition to sensible intuition, and sometimes they were limited by phenomenalism. Yet, Husserl’s platonist epistemology is powerful precisely because he can also posit categorial intuition and eidetic intuition. These are non-mystical qualities of understanding which we use in every-day life, and are the basis for our mathematical knowledge at every level.

Naturalists today are still essence-phobic. Some of that phobia is justified, but not when it comes to formal sciences. Even eminent minds like W. V. O. Quine criticizes the platonist understanding of meanings, saying that "meanings are what essences become when they are detached from the object and wedded to the word". Yet, his epistemology limits itself to the positing of mathematics and logic as indispensable to science itself, yet, he has some problems:

  • He is unable to explain why logic and mathematics are necessary for science. He just argues, from the pragmatic point of view, that they work.
  • He is unable to explain the success of the predictions made by logic and mathematics, especially in developments which initially seem pointless: as it happened with non-euclidean geometry, negative roots, Hilbertian spaces, among other mathematical developments.

Paul Benacerraf, in his famous essay "Mathematical Truth", states that for a mathematical proposal to be acceptable, it must account for the objectivity of mathematical truths (something which platonism does very well), and the knowledge of mathematical concepts (which he felt platonism couldn’t accomplish). Husserlian platonism fulfills both requirements very, very well.

Like the Bible says: "By their fruits, ye shall know them."

In essence, what Husserl gives us is the reason we can call this a genuine platonist epistemology. We know formal structures from our experience when we constitute states-of-affairs, yet, when we carry out formalization (categorial abstraction), there is no trace of sensible objects anywhere, and we are also able to intuit the necessary and universal relations among these mathematical objects.


Benacerraf, P. (1983). Mathematical truth. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics. (pp. 403-420). UK: Cambridge University Press.

Brown, J. R. (1999). Philosophy of mathematics: an introdution to the world of proos and pictures. London & NY: Routledge.

Hill, C. O., & Rosado, G. E. (2000). Husserl or Frege? Meaning, objectivity and mathematics. IL: Open Court.

Husserl, E. (1973). Experience and judgment. (J. S. Churchill & K. Ameriks, Trans.). London: Routledge & Kegan Paul. (Original work published in 1939).

Husserl, E. (1998). Ideas pertaining to a pure phenomenology and to a phenomenological philosophy. The Hague: Kluwer Academic Publishers. (Originally published in 1913).

Katz, J. (1998). Realistic Rationalism. Cambridge: The MIT Press.

Husserl, E. (2001). Logical investigations. (Vols. 1-2). (J. N. Findlay , Trans.) NY: Humanities Press. (Original work published 1900/1901, 2nd ed. 1913).

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 — 5

On April 16, 2011, in Philosophy, by prosario2000


One of the things I’ve learned since I began studying philosophy is that our theoretical framework determines what we observe, what we see. One of the examples I learned this from was in a very good anthology of readings on Gottlob Frege, edited by Michael Beaney. I highly recommend it, Beaney does a great job in compiling essential readings of Frege, as well as some translations made by Beaney himself.

The Frege Reader

The only thing I don’t like about it is his refusal to translate the word Bedeutung. Beaney’s attitude is understandable, Frege made a very poor choice of words regarding that term, since the word Bedeutung in German usually means "meaning" (pardoning the redundancy), while Frege uses it to mean "referent" or "denotation". In Logical Investigations, Husserl quotes Frege only two times in all of that magna opus, and in one of those occasions was precisely to criticize Frege’s use of the word Bedeutung in this particular way (see Inv. I, § 15). So, I understand Beaney’s refusal to keep the German word Bedeutung, but if by that, Frege means "referent" … then I think it should be translated as "referent".

The book has the received view originating from an essay by Dagfinn Føllesdal, a renowned philosopher and W. V. O. Quine’s disciple. This essay was called "Husserl’s Notion of Noema" published in the Journal of Philosophy in 1969. In it, Føllesdal states that Husserl converted to platonism and left psychologism as a direct result of Frege’s review, something which, by this stage of our series we know to be false. That is not the only allegation, though. He also says that when we look at Husserl’s notion of "noema" in his phenomenological stage, we must realize that this notion is nothing more than a generalized version of Frege’s "senses" (Sinn). Husserl made the distinction between sense and referent in Logical Investigations, which was written in 1900, clearly after Frege’s review in 1894, and Frege’s famous masterpiece "On Sense and Referent" (1892). And for Husserl, a noema is nothing more than an a sense (meaning) given by an intentional act. Hence, a noema is the sense and the referent is an object. Right? So, Husserl apparently became a platonist a la Frege and his contribution is the generalization Frege’s senses to the realm of presentations. Without Frege, no phenomenology (at least a la Husserl). Interesting!

As I said, Beaney’s book apparently subscribes to this view originated by Føllesdal’s essay (Beaney, 1997, p. 6). Part of what makes Beaney’s anthology so good, though, is that he includes in it a portion of Frege’s letter to Husserl dated May 25, 1891. This letter was a response to Husserl sending Frege his Philosophy of Arithmetic, and his review on E. Schröder’s work on mathematics. In it, Frege says the following (I’ll translate "Bedeutung" as "referent"):

Dear Doctor,

I thank you especially for your Philosophy of Arithmetic, in which you take notice of my own similar endeavours, perhaps more thoroughly than has been done up to now. I hope to find some time soon to reply to your objections. All I should like to say about it now is that there seems to be a difference of opinion between us on how a concept-word (common name) is related to objects. The following schema should make my view clear:

Frege's Theory of Sense and Referent

With a concept word it takes one more step to reach the object than with a proper name, and the last step may be missing — i.e., the concept may be empty — without the concept word’s ceasing to be scientifically useful. I have drawn the last step from concept to object horizontally in order to indicate that it takes place on the same level, that objects and concepts have the same objectivity. . . . Now it seems to me that for you the schema would look like this:

Husserl's Conception of Concept-Word

so that for you it would take the same number of steps to get from proper names to objects as from concept-words. The only difference between proper names and concept-words would then be that the former could refer to only one object and the latter to more than one. (Beaney, 1997, pp. 149-150).

SEE???!!!! Right THERE!!!!! … Beaney apparently didn’t see it in his own anthology, but it’s there!!! Get it???!!!!

If you didn’t get it, let me spell it out for you. Føllesdal argued in 1969 that Husserl made the difference between sense and referent as a result of Frege’s essay "On Sense and Referent" (1892), and Frege’s review (1894). Yet, here, in a letter written in 1891(!!!!), Frege says clearly, that Husserl had made the distinction between sense and referent by that time. Both men seem to have come up with the distinction of sense and referent simultaneously, but independently … much like the way Newton and Leibniz both developed calculus.

This distinction did not appear in Frege before 1890, but this distinction did appear in Husserl’s review on Schröder’s work, a review published in 1891. Føllesdal was wrong, once again. J. N. Mohanty, Claire Ortiz Hill, and Guillermo E. Rosado Haddock have worked extensively on this subject of Husserl’s development of his semantic doctrine. Claire Hill (2001) and Jaakko Hintikka (1995) have worked also in clarifying the relationship between Husserl’s notion of noema, and Frege’s notion of sense. Below I give you the references if you wish to look for them.

For now, I wish to ask the question. How come two very different philosophers come up with almost the same distinction? The answer lies in the mathematical and semantic studies of a relatively unknown Czech priest at the time called Bernard Bolzano, whose work Husserl and Frege knew very well. I’ve said in a previous blog post that Husserl’s reading of Bolzano’s works helped him turn away from psychologism. Well … here is a very brief version of his story and why he influenced Husserl the way he did.

A Priest and the Birth of Semantics

Bernard Bolzano

Born in Prague in 1781, Bernard Bolzano studied mathematics, philosophy, and physics in the University of Prague. Due to his dedication and devotion to God and the Catholic Church, he studied theology and was ordained a priest in 1804, and later appointed to be a position in philosophy of religion. Due to political problems and his stance for peace, he was dismissed from the university in 1819, and he spent the rest of his life writing on many subjects, including mathematical and philosophical. He never had too much exposure mostly because he was forbidden to publish in mainstream journals as a condition for his exile. He had no choice but to publicize his ideas in other sorts of journals and publishing businesses, not as widely circulated. He died much later in 1842.

Despite the obscurity of such publishing methods, his writings reached the hands of Franz Brentano, Edmund Husserl, Georg Cantor, and others, such as Gottlob Frege. In fact, Husserl first heard of Bolzano in one of Brentano’s seminars. He read him, and although very impressed by his philosophy, he felt pretty much uncomfortable with some "objective, existent, abstract" beings posited by Bolzano, which for Brentano were nothing more than fictions. As a mathematician and a proponent of psychologism (that is, before 1890), Husserl found it difficult to deal with Bolzano’s philosophy.

One of the most important works by Bolzano was Theory of Science, where he pondered a lot about some issues which seemed to arise from mathematics. One of them has to do with the core issue of all philosophy of mathematics: "what the he** is a number?" One thing is to represent a number this way "2", or this way "II" or this way "two". These different representations are obviously just signs, i.e. physical expressions which stand for something else. Yet, numbers don’t seem to be physical objects either, because in mathematics we never really deal with propositions like "two oranges and two other oranges make up four oranges", but "2+2=4". We deal with numbers in their purity from all physical content. How do we deal with this?

Bolzano distinguished between what he called "subjective representations" and "objective representations" (if this sounds like the Husserl of Philosophy of Arithmetic, it is no accident). Many people can have a subjective image or representation of their own minds, yet it seems as if despite the differences in our representations, there is a general agreement of something objective we do seem to share. If I show you a picture of the phone, each one of you will see it in a different manner, because sight depends on perspectives, where the light hits, if you are color-blind, and so on. Yet we all understand what a phone is in an objective manner despite our different mental ssubjective representations.

In the same way, we can represent numbers with different physical signs, either through symbols in writing or the sound when we pronounce the words for the number two in different languages ("two", "dos", "deux", "zwei", etc.) Yet, despite these psychological or physical differences with the way we represent ourselves the number two, there is a sort of objectivity we share. That is the objective representation we somehow understand (but do not represent in our own imagination) in our minds.

Bolzano says something interesting:

[Objective representations] are not to be found in the realm of the real. An objective representation does not require a subject but subsists, not indeed as something existing, but as a certain something even though no thinking being may have it; also, it is not multiplied when it is thought by one, two, three, or more beings. . . . Fort his reason, any word, unless it is ambiguous, designates only one objective representation (Coffa, 1991, p. 30).

By not existing, what Bolzano meant is that "objective representations" are not in space or time as all physical objects are, yet subsist in some way or manner. They are different from mental representations, and at the same time the object (referent) of that representation.

He also talked about propositions, that is, those sentences which can be said to be true or false: "The grass is green", "The car is yellow", "The computer screen is not working". There are subjective representations of these propositions, the physical ink and the particular letters, or whenever you hear the sounds when I say these things.

Yet, there is also an objective side of propositions, these he would call "propositions-in-themselves" (Sätze an sich), which is made up of objective representations and they are related through copulae (and, or, if … then, and so on). So, if I say that "the car is yellow", the words "car" and "yellow" express an objective representation of both a car and the color yellow, but joined together with the word "is". With the word "is", we have linked the two (the car and the color yellow) in a subject-predicate manner. In essence, for all practical purposes, Bolzano is developing a theory of grammar, so that propositions make sense (or have meaning). This is precisely what semantics is all about (theory of meaning).

Unfortunately Bolzano nor Frege or Husserl used the word "semantics" to describe it. It was still called "logic" at the time. But, one thing that this theory led is to a doctrine of analytic and synthetic judgments. Coffa (1991) sums up very well Bolzano’s point using two examples:

  1. This man is a featherless biped.
  2. If all men are mortal and all Greeks are men, then all Greeks are mortal.

The difference between both propositions is that the first one is a posteriori (which means that it has to be verified by experience). The truth of proposition (1) will inherently depend on whether that particular man is a featherless biped.

We cannot say the same thing about proposition (2). It seems as if the truth of this proposition did not actually depend on the existence of men, or Greeks, or even mortal beings. The absolute truth depends on the grammatical arrangement, or the form of the proposition. To give you an example:

"If all teavies are born in Paris, and if Paris is Honduras’ capital, then all teavies are born in Hondura’s capital".

And this is absolutely true despite the fact that no one knows what "teavies" are, and that Paris is not Honduras capital. The grammatical structure "if … and if … then …" seems to make all the difference. Eventually, Husserl will adopt this view to elaborate his own doctrine on analytic and synthetic judgments.

Husserl and Bolzano

Husserl considers his views as being a more sophisticaded and elaborated version of Bolzano’s ideas. Bolzano’s use of the term "representations" (Vorstellungen) which is a psychological term, to refer to two different things: one subjective, the other objective. Husserl’s error when he was a psychologist, was to believe that these objective representations are in some sense psychological in origin. Bolzano, clearly, did not think that way, and he was right.

Despite Bolzanos genius, his use of such a language did not let him reach the final difference between sense and referent. One is a bit disappointed, because it is watching a tree whose fruits are all ripe and mature, and all you have to do is to shake the tree a bit so that the fruits finally fall. Husserl and Frege, independently, were able to finally make the distinction, which was hidden in Bolzano’s misguided use of the term "objective representation" to refer to nothing more than sense or meaning.

As we shall see here, as in the case of Bolzano’s "objective representations", senses or meanings are themselves totally abstract but objective. They are not found in space or time, but in a sense subsist independently of the mind or the physical world. They are not the physical sign, nor do they belong in the psychological activity of the brain. If I say: "Kennedy was killed in 1963", its meaning, its proposition, is true, and will always be true, even if everyone believed it to be false. There is no better example to show that senses or meanings are independent of the activities of the mind. Psychologism’s error was to believe that they were not independent. As we shall see in a future blog post, this will lead to inevitable contradictions.

Husserl semantic doctrine, a doctrine of meaning is indeed exactly as Frege described in his 1891 letter, and Husserl will restate his semantic doctrine in his Logical Investigations (Inv. I. § 12).

Sense (Meaning) and Referent of Proper Names

For Husserl, there are two sorts of names: a proper name or a universal name. For Husserl (as for Frege), a proper name is a word or phrase which is used to refer to one single object. It could be a strict proper name such as "Aristotle", "Lisa DeBenedictis" or "Martin Luther King", or what Bertrand Russell would call "definite descriptions" or "denoting phrases" such as: "the king of France".

Let us use the following examples to illustrate the difference between proper names, their sense (meaning), and their referent (I’m going to use a mix of Frege’s and Husserl’s examples):

Example 1:

"The victor in Jena"

"The victor in Jena"

Both proper names are exactly the same, because we are using the same sign or physical represenation in graphic writing. Hence, they contain the same message (sense or meaning), and refer to one and the same object: Napoleon Bonaparte.

Example 2:

"The victor in Jena"

"El vencedor en Jena"

Here both proper names differ, because the graphical and written representations or signs are different. One of the phrases is in English, the other one is in Spanish. Yet they both carry the same sense or meaning, because they tell us the same thing in both languages. They still refer to the same object: Napoleon Bonaparte.

Example 3:

"The victor in Jena"

"The defeated in Waterloo"

Here, both proper names differ, but there is now a difference in sense or meaning. They both refer to Napoleon Bonaparte, but not in the same way. Why? Because both of these proper names are giving us two very different informations about the guy. Since their logical content is different, their sense or meaning is different.

The same is true when we have other sorts of proper names which give us different abstract content (sense or meaning) but referring to the same object: "the equilateral triangle" and "the equiangular triangle"; or, to use Frege’s own example, "the morning star" and "the evening star" to refer to planet Venus.

Example 4:

"The morning star"

"The defeated in Waterloo"

Both proper names are different, both senses or meanings are different, and they both refer to two different objects: the first refers to Venus, and the other one Napoleon.

So, we can sum up these examples in the following way in this table:

Sense-Reference Table

Sense (Meaning) and Referent of Universal Names

The theory of sense (meaning) and referent of universal names is exactly as Frege described above. For Husserl, a universal name is whatever names a set of objects. For example, the term "horse" is a universal name. Its sense or meaning is a concept, while the referent are all of those objects which fall under that concept (or extension of the concept).

For example, the universal name "horse" expresses a concept which can have a variety of objects: Black Beauty, Rocinante, Napoleon’s white horse, and all of the horses which have existed and exist today. This works out pretty well in today’s semantics, and it has been adopted in contemporary philosophy.

On the other hand, Frege’s version was left behind. Because of his logicist proposal, and his notion of the concept as a logical function of one argument, he had to place the concept itself at the level of referent (as he graphically explained in Husserl’s letter). Yet, as Guillermo Rosado Haddock has pointed out, in his many of later works, Frege never clarifies what he means or how to exemplify the sense of a concept-word, hence leaving a big hole in his semantic doctrine.


But what about "propositions"???!!! We will talk about it in our next blog post.


Bernet, R., Kern, I., & Marbach, E. (1999). An introduction to Husserlian phenomenology. IL: Northwestern University Press.

Beaney, M (ed.). (1997). The Frege reader. US: Blackwell.

Coffa, J. A. (1991). The semantic tradition from Kant to Carnap: to the Vienna Station. US: Cambridge University Press.

Føllesdal, D. (1969). Husserl’s notion of noema. Journal of Philosophy, 66, 680-687.

Hill, C. O. (2001). Word and object in Husserl, Frege, and Russell: the roots of twentieth century philosophy. US: Ohio University Press.

Hill, C. O. & Rosado Haddock, G. E. (2000). Husserl or Frege? Meaning, objectivity, and mathematics. US: Open Court.

Hintikka, J. (1995). The phenomenological dimension. In B. Smith & D. W. Smith, The Cambridge companion to Husserl. US: Cambridge University Press.

Husserl, E. (2001). Logical investigations. (2 vols.) London & NY: Routledge.

Mohanty, J. N. (1974). Husserl and Frege: a new look at their relationship. Research in Phenomenology, 4, 51-62

Mohanty, J. N. (1982a). Edmund Husserl’s theory of meaning. The Hague: Martinus Nijhoff.

Mohanty, J. N. (1982b). Husserl and Frege. IN: Indiana University Press.

Rosado Haddock, G. E. (2006). A critical introduction to the philosophy of Gottlob Frege. US: Ashgate.

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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