Texto introductorio sobre Edmund Husserl (v. 0.6)

On August 8, 2014, in Philosophy, by prosario2000

Para todos aquellos interesados en la filosofía de Husserl, he dedicado gran parte del verano a escribir. Uno de esos escritos es un libro de texto sobre Husserl. Nótese que no es la versión 1.0 (que estará disponible para diciembre de este año).

Todavía las referencias son incompletas y, a pesar de considerables correcciones, es posible que se me haya colado uno que otro error. Agradezco cualquier señalamiento o crítica constructiva del escrito. Pueden escribirme a prosario2000@gmail.com.

Pueden descargar el documento aquí.

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Some Points about Process Philosophy

One of the things I have been looking at recently is the metaphysics proposed by A. N. Whitehead, and many of its variants, especially those proposed by John B. Cobb, Jr. and David Ray Griffin (the latter was one of the most recent editors of Whitehead’s Process and Reality). I want to question some of its central aspects of this philosophy in light of several issues related to current discussions of mathematical platonism.

My platonist framework is that of Husserl, who conceived categorial logical forms as ideal structures, and not as in the case of Gottlob Frege as “objects” (in the sense of saturated entities). As it happens, just as process philosophy’s emphasis on relationality, these logical forms are conceived as relational.  Yet, these logical relations are not themselves in process, hence, not actual.

Process Philosophy’s Criticisms to Aristotelian Metaphysics

Aristotelian philosophy is wholly based on the notion of substance, which has been problematic for many philosophers for ages. Its metaphysical standing has been disputed by many for being inadequate as a conceptual basis for science.

According to Aristotle, we can distinguish substance from attributes. A substance is, by definition, something that is self-subsistent, while an attribute is a property of a substance. A horse is a substance, a chair is a substance too. They can be conceived as existing with independence from other objects. Yet, the color blue cannot be a substance. Have you seen a color blue just floating in the air?  No. It is necessarily an attribute of a substance, it cannot exist without a substance. Given this, a substance can be conceived as the subtract of all the attributes of an object.

For Aristotle, in every substance we can find matter and form. Matter is whatever the substance is made of, but form is what defines what the substance is. For example, a chair is made out of wood, that is its matter (but it doesn’t define what the chair is). Yet, the form (the shape or the specific material arrangement of matter) that the wood takes in order to function is what defines a chair as a chair. Aristotle calls “essence” what a substance is (being a chair is precisely the essence of the substance we are talking about, and its form (among other things) determines its essence). The object itself was called by Aristotle “primary substance“, while its essence (the kind of thing that the primary substance is) would be the “secondary substance“.

For him, we can conceive a substance as unchanging, while its attributes do change. For example, if I have a green chair, and I paint it yellow, it continues to be one and the same chair, while its color attribute has changed. By changing its color, the chair has not stopped being what it is (its essence is unaltered). But if I take the wooden chair, burn it, and it becomes ashes, the chair has ceased to be what it was, to become something else (ashes).

Finally, for Aristotle, states that relations themselves are founded on substances. Substances are primordial, while relations are secondary.

Process philosophers challenge the very notion of substance. Their objections are not new, and are presented today as the most convincing arguments against this sort of metaphysics. They point at the way some Modern Idealists criticized the whole idea of the notion of substance as being primordial. Wildman (2013) gives us the example of a keychain within a given relational context. What makes the key what it is? Its essence, so-to-speak, is derived, not from its shape or form, but its relational context: Its essence changes, when the relational context changes: if we change all of the locks, the purpose of the keychain changes completely –before it was to open the lock, now they are useless–. They lost the essence of being keys (i.e. to open locks) (Wildman, 2013, p. 86). As process philosophers have pointed out, in this case, we can see clearly that the relational aspect is primordial. From this perspective, the idea of a substance or essence that does not change while attributes change become incomprehensible.

Process Metaphysics

Whitehead, Cobb, Griffin, and the rest of process metaphysicians, establish relationality and not substance, as the primordial factor of their metaphysics. According to them, the primary entities of the world are not objects (saturated entities), but what they call actual occasions or occasions of experience. These terms are the primordial concepts with which they describe change (in the Heraclitean sense). The way they conceive actual occasions is like a series of droplets of occasions that constitute temporality. The future is open-ended (there is no future at all), the past occurred already (but in a sense, endures), and the present is a set of actual occasions that become, change, and pass. (Note: I find this somewhat problematic from a scientific standpoint, especially in light of special and general relativity. I will elaborate this problem further in another post.)

I wish to point out something that is perhaps susceptible to misunderstandings regarding process. The fact that change is the rule in the physical world does not mean that literally everything real changes. In fact, process philosophy recognizes a level of nomic permanence, something that reminds us of the notion of logos as conceived by Heraclitus and the Stoics. In the words of John Cobb, Jr.:

Process thought … does not assert that everything is in process; for that would mean that even the fact that things are in process is subject to change. There are unchanging principles of process and abstract forms. But to be actual is to be a process. Anything which is not a process is an abstraction from process, not a full-fledged actuality (Cobb & Griffin, 1976, p. 14).

This statement is very important, because we are going to discuss some aspects of this nomic aspect proposed by process, and the issues raised by a platonist-structuralist view of mathematics.

Finally, I wish to mention the fact that for process philosophy, no individual is a strict object (in Fregean language, a saturated entity). Each unit of change can be described as an individual, but every individual is in itself a society of individuals interrelated with each other organically. Each individual has an inner reality, as well as an external interconnectivity with other individuals. In this sense, process thought conceives relationality “all the way down”, each individual is made up of an interactivity of individuals, which simultaneously are made up of further interactive individuals, and so on. A cell is an individual, but a rock is not. A rock is made up of individual molecules, but it is not organized in such a way as for the rock to relate organically with individuals.  Each human is nothing but a society of individuals. As it turns out, from a strictly biological standpoint, this is exactly true!

Husserl’s View of Categorial Forms as Ideal Relations

I would like to point out some aspects of Husserl’s philosophy that are pertinent in this discussion. He made the distinction between relations-of-ideas and matters-of-fact, a distinction inspired by David Hume. In the realm of relations-of-ideas, we can include logico-mathematical relations, as well as essences themselves (by “essences”, Husserl refers to a certain sort of meanings, the concepts). All true judgments about relations-of-ideas (i.e. truths-of-reason) are:

  1. Analytic-a priori statements: Which include analytic laws, statements devoid of all material concepts that are always true; and analytic necessities, particular instances of analytic laws. For instance: a + b = b + a would be an analytic law, but the statement “two apples and one apple always is an apple plus two apples”, expresses an analytic necessity.
  2. Synthetic-a priori statements: These are always true, but they cannot be formalized salva veritate. For example, the statement “no color can exist without a colored surface”.

Analytic and synthetic-a priori statements are true due to the essence of what they propose. However, all true statements referring to matters-of-fact (i.e. truths-of-fact) are synthetic-a posteriori.

Since all analytic and synthetic-a priori statements are necessarily true, then that means that they establish a ideal nomic realm (i.e. “laws” that will rule this realm out of logical necessity). In the case of the analytic-a priori statements, we are given logical-mathematical truths, what Husserl called a mathesis universalis.

According to Husserl, in all judgments we can identify formal components that relate other concepts and other judgments. These formal components are called by him “meaning categories“, which are ruled by laws that we today call “rules of formations” (laws to prevent non-sense or meaningless judgments), and “rules of transformation” (laws to prevent contradictions). None of the analytic laws in this realm refer to anything to matters-of-fact, nor are they “actual” (in the process sense of the term). Some of the meaning categories include: Subject-predicate structure, conjunction, disjunction, forms of plural, forms of combining new propositions from simpler ones, and so on. In this sense, formal logic is formal theory of judgment or formal apophantics.

Judgments or propositions refer to states-of-affairs (facts), i.e. sensible objects arranged in a specific ideal manner. These ideal arrangements or ideal relations of objects are called by Husserl, “formal-objectual categories” or “formal-ontological categories“. These elementary forms of arrangements of objects can be: unity, plurality, sets, cardinal number, ordinal number, part, whole, relation, among others. Mathematics develops a theory based on each of these formal-objectual categories: for instance, from the category of sets we develop set theory; from the category of cardinal numbers, we can develop an arithmetic of cardinal number; from the correlative concepts of parts and wholes, we can develop a mereology, and so on.

Husserl’s mathematical Realism and Platonism is evident once we realize that he ontologized these categories. For him, the concepts of cardinal numbers, sets, parts and wholes, etc. refer to ideal and self-subsisting entities. This is due to the fact that mathematical statements being necessarily true, must refer to these existent ideal structures in order to be true (this is now called the “ontological commitment” factor in mathematics). Not only that, but these formal-ontological categories can themselves become objects of still higher abstract states of affairs. For example, If I have, the set {A,B}, I can include as element another set to create a set of a higher order {{A,B},{C,D}}, in which case, the sets {A,B} and {C,D} become objects (elements) of the set of a higher order. We can study these formal-ontological categories and their theories completely devoid of all sensible or material components (essentially by substituting objects with variables). In this way, mathematics is a formal theory of object or formal ontology: it studies the forms of being of any object whatever!

Both, formal apophantics and formal ontology form together a mathesis universalis, what Leibniz considered the supreme form of mathematics.

Critical Evaluation of Process Thought in Light of Husserl’s Platonist Philosophy

Notice that in each of the cases, judgments and states of affairs, we can distinguish between “matter” and “form”. In the case of states-of-affairs, sensible objects constitute the matter, while the formal-objectual categories are the form in which they are arranged. In a sense, they are their own unit constituted in the world, and as such, they can be experienced (phenomenologically speaking).

This has important consequences for process. For this way of viewing reality, the world is made up essentially from occasions of experience. We can say the same thing from a Husserlian standpoint. After all, for Husserl, the “world” or the “universe” is the sum of all existent objectualities (states-of-affairs) in the temporal realm, which is the correlated with the deductive network of truths-of-fact. Neither Husserl nor process philosophers conceive the world as made up of mere objects, but of the way these objects are related. There is a difference between Husserl and process, though. For example, every individual is in a relational organic relationship with other individuals. Husserlian states-of-affairs are specific ideal arrangements of objects (any sort of objects, not just individuals). The rock forms part of the set of objects in the garden, but it is not an individual in terms of process, nor does it have the sort of arrangement that would make it have an “inner life” so-to-speak.

In this sense, the organic relations among individuals are nothing more than a subset of possible formal-objectual abstract relations that objects have. In this sense, individuals are nothing but an organic whole made of many sorts of states-of-affairs, all objectually arranged and based organically on one another “all the way down” … but not “infinitely down”. Some relations are clearly based on ultimate subtracts. This can be seen more clearly in the debate between Platonism and Structuralism. For Structuralism, abstract relations are primordial over the mathematical entities themselves, mathematical objects are defined by the places they fill within a given structure. However, when you look at sets, these forms make sense as relations based on any objects whatever. If this is the case, structuralism cannot give us an account for sets at all (Brown, 1999, pp. 62-66). We find the same defect in process.

Considering this way of viewing things, from a Platonist-Husserlian standpoint, problems begin to emerge for process philosophy. For instance, if organic relations are nothing but subsets of all possible formal-objectual relations of given objects, then by essence relations cannot be the “most” fundamental components of occasions of experience. On the contrary, Husserl’s views on states-of-affairs seem to be closer to Aristotle than to Whitehead. If there are relations in the realm of matters-of-fact, there must be subtracts that become the objectual elements of those relations. It would be non-sense to insist that relations are primordial, since the ultimate elements of formal-objectual categories must be objects (in this case, material). There cannot be a second-order set without at least a first-order set. There cannot be any relation among individuals in the physical world if the individual is not itself a physical object (in the broad sense of the term “object”). The objectual relationships occur from bottom-up, not from up to bottom. So, in a very real sense, individuals cannot be made up of “societies all of the way down” (at least not literally). It can only go down to a point, and that point must be in some sense substantial.

Alternative Proposal: Nuanced Conception of Substance (Object) and Relationality

Since it is clear that the Aristotelian notion of substance (and even the most radical conception –the Cartesian version–) is not adequate, nor is the process version either, then we must choose a better alternative, one I rarely see discussed: the notion that substance (object) should be co-fundamental with the one of relations.

We can talk about objects and their formal relationships in a state-of-affairs. Whitehead said that our experience (subjective and objective) is of occasions of experience. Here I want to suggest a return to Husserlian phenomenology and recognize that what is experienced are states-of-affairs as such, and that objects and their formal relations are given to any consciousness (any ego) simultaneously or any other sort of individual. Not only do objects arrange themselves according to form according to ideal laws (that are ideal, necessary, and unchanging), but that is the only way we perceive and know them. Only in this way, we can know and experience certain kinds of states-of-affairs that we can term “occasions of experience” (in the realm of temporality), and we can understand ourselves as individuals in the process sense of being a hierarchy of relational networks of individuals. No object would be understood necessarily as a Fregean saturaded object.These objects can be space-time, superstrings, forms of energy, material objects, entangled quanta, quarks, and so on. However, we should keep in mind that to suppose an “infinity” of relationality “all the way down” begs the question. For this reason, we can also open the possibility of foundational objects (in the general sense) that are not constituted by a further relationality of actual occasions, and that, despite this, are themselves in a relationship with other objects.


Brown, J. R. (2008). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. London: Routledge.

Cobb Jr., J. B. & Griffin, D. R. (1976). Process Theology: An Introductory Exposition. Louisville: Westminster John Knox Press.

Husserl, E. (2008). Logical Investigations. London & New York: Routledge.

Rosado Haddock, G. E. (2003). 14. On Husserl’s Distinction between State of Affairs (Sachverhalt) and Situation of Affairs (Sachlage). In Husserl or Frege? Meaning, Objectivity, and Mathematics. C. O. Hill & G. E. Rosado Haddock (eds.). pp. 253-262. IL: Open Court.

Whitehead, A. N. (1978). Process and Reality. NY: Free Press.

Wildman, W. (2013). Una introducción a la ontología relacional.  In La Trinidad y un mundo entrelazado. J. Polkinghorne (ed.). pp. 81-102. Navarra, España: Editorial Verbo Divino.


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A Model of Philosophy for Philosophers Everywhere …

On November 3, 2013, in Philosophy, by prosario2000

Logical InvestigationsInvestigaciones Lógicas


I am currently writing a book regarding science, ethics, and society. It will take some years to finish (mainly for being a big and ambitious project). For the time being, I need to build a suitable metaphysics that is coherent ans logically consistent. For that task, I decided to use Edmund Husserl’s own views on logic and mathematics, and simultaneously see if I could use some ideas from process philosophy as t was developed by A. N. Whitehead in Process and Reality, and other philosophers and theologians.

As a starting point, I began studying Husserl’s change of mind from a psychologistic to a realist and platonist conception of mathematics and logic. Then I started reading again Husserl’s major masterpiece Logical Investigations. The first volume of this work is called “Prolegomena of Pure Logic”, a set of eleven chapters dedicated to provide a proper understanding of logic. This is the book that turned me into a platonist, and it never ceases to amaze me. This is one of those texts that you read time and again, and discover something new. What I particularly love about this part of Logical Investigations is that it has all of what it means to be a good philosopher writing a great philosophical work.

Most analytic philosophers deal with Gottlob Frege’s refutation of psychologism in his masterpiece The Foundations of Arithmetic, where he argued against psychologism, naturalism, formalism, fictionism, and other forms of antirealism. Yet, how ever great and neat are some of his arguments, he had two vices. The first was to distort (perhaps inadvertedly) or purposely mock his opponents’ opinions so that his looked greater and better. This tendency is clearly seen in the famous 1894 abusive review of Husserl’s psychologistic work Philosophy of Arithmetic, published in 1891 (before Logical Investigations). The second flaw of Frege’s arguments is that often he was very brief in his assessment of his opponents’ views. For instance, you see very good arguments against psychologism in “Thought: A Logical Inquiry”, or in his posthumous published articles on logic.

On the other hand, Husserl was, in this sense, a very different deal. This is in part because he was Franz Brentano’s disciple, and held a mild version of psychologism himself for a while (see his works published from 1887 to 1891). This means that he really understood the different philosophers most read in his time, and he was taught by one of the most prominent psychologists in the field. Yet, Husserl was also a mathematician by training. He was Karl Weierstrass’ disciple and assistant, and took some courses with Leopold Kronecker, and Felix Klein. Georg Cantor, who was his mentor and later one of his closest friends, was part of the panel when he defended his Habilitationsschrift, “On the Concept of Number”. Each of these characters were the crème de la crème of the mathematical revolutionaries of the nineteenth century (needless to say that he later befriended Hermann Weyl, Ernst Zermelo and David Hilbert; during 1901 to 1916, Husserl belonged to Hilbert’s Circle). It was the mathematical side of him that made him change his mind away from Brentano’s teaching towards a different notion of logic.

Why is the “Prolegomena” superior, so much superior, than Frege’s criticisms, and why do I consider it as a model for what philosophical works should look like? In one aspect, Frege and this part of Husserl’s work share the first virtue: they are written clearly and with precision regarding terminology and exposition. Their honesty permeates their work. However, Frege and Husserl begin to depart in a second very important aspect. Frege was brief when criticizing psychologism in many of his works, yet, Husserl dedicates about nine whole chapters to refute psychologism. He is both methodical and extremely thorough, even to the point where he makes very important distinctions among different sorts of psychologism, distinctions that Frege never cared to make. This led the latter to assert that all psychologists (those who held a psychologistic point of view) fell into individual relativism, i.e. the assertion that all truths are relative to individuals. Meanwhile, Husserl did establish a difference between those who fell in individual relativism, and what most psychologists actually did fall into: specific relativism, i.e. that truth is relative to the structure of a species (in this case, the human species).

The way he developed his criticism to psychologism is well-thought out, and very organized:

  • Defining the problem: whether logic is technique (“tool for rightful reasoning”) or theoretical (that tells us the formal a priori syntactic and deductive relationships among concepts and propositions respectively, regardless of what anyone thinks). If it is the former, then logic should be considered a branch of psychology (which is what psychologism proposes) and purely normative in nature (i.e. it tells us how we ought to think). If it is the latter, then logic is its own field apart from psychology (and tells us what is) (Chapter 1)
  • The idea of theory of science (knowledge) and how logic relates to it (Chapter 2)
  • Theoretical truths as basis for normative disciplines (or how is the normative side of logic founded on the theoretical one) (Chapter 2)
  • The arguments of psychologism, and why they have been usually triumphant in relationship with the arguments presented by antipsychologists (Chapter 3)
  • The empirical consequences of psychologism and their respective refutations (Chapter 4)
  • Refutation of John Stuart Mill’s and others’ psychological account of logical knowledge and validity (Chapter 5)
  • Refutation of psychologistic conceptions of syllogistics, especially in relationhip with physical and chemical laws (Chapter 6)
  • The skeptical relativistic consequences of psychologism and how they can’t be supported (Chapter 7)
  • The refutation of the psychologistic prejudices found beneath all of the psychologistic arguments (Chapter 8)
  • Anthropological view of logic (principle of the economy of thought) and its refutation (Chapter 9)

Some people have described the “Prolegomena” as presenting the most memorable refutation of psychologism, and, to this day, this philosophical view has had a very hard time recovering.

The reason why it was so amazingly successful is the whole point of this blog post. It is a work, not filled with arrogance, but instead of intellectual humility. He granted his opponents all of their own assertions without trying to distort them for his own sake. He even praised a lot of their opinions, especially whenever he found them worthy to be noticed. Yet, he prepared his argument with simple, clear, and evident statements that clarified his position to then see why they fail miserably, even granting them all of the fairness in the universe.

His Cartesian spirit of trying to provide a philosophy on clear and evident thinking made Husserl go a long way (in many ways still not widely recognized today). He presented the most convincing epistemology of mathematics to date, he sowed the seed to create the field of mereology (his third logical investigation considers the matter of “parts and wholes”), he proposed the nomological-deductive scheme of physical theories (about 30 years before Popper and Hempel came up with it) and further elaborated on the underdetermining aspect of scientific theories, he recognized a universal grammar as a necessary aspect of any language that refers to facts in general (decades before Noam Chomsky’s proposal), he further recognized the validity of non-Euclidean geometry two decades before Einstein’s general theory of relativity, he distinguished between rules of formation and rules of transformation (using different terminology), formulated of his theory of manifolds as the closest description to actual practice of logic and mathematics at that time and today, his phenomenological work was in many ways a basis for psychological research (like the Gestalt school of psychology) back then and today, and much more.

If you want to be a great philosopher, be humble and honest (especially with yourself). Test and evaluate your thinking every single day of your life. Intellectual humility goes a long way, farther than people think. Husserl is a perfect example of this. No wonder he should be considered the greatest philosopher of the twentieth century.

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:

Old Website

(Click for Larger Version)

This is how it looks like:

Website in Browser
(Click for Larger Version)

You can look at the new website by going to http://uos.pmrb.net. I hope you like it. Any comments or questions about it, please, let me know.

The book is completely available online under different formats. You can download it for free and copy it as many times as you wish just under two conditions: the original work will be preserved verbatim, and no commercial use of it is allowed unless you have reached an agreement with me. Additional to this, because the cover is a derived copylefted version of a GPLed wallpaper in KDE-Look.org, I released the cover and all of its new graphic elements under the GNU GPL as well, and allow people to download it and use it as they wish commercially or non-commercially as long as they comply with that license.

The book is also available for sale for now in Lulu.com.

I hope that this book will help contribute to a clearer understanding about the nature and role of formal sciences such as logic and mathematics, and natural sciences such as physics and biology.

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Edmund Husserl

Up to now, we have seen that psychologism (i.e. conceiving logic as a normative discipline which tells us how to think, or reducing logic to matters-of-fact especially regarding the mind) is unfruitful and implies all sorts of counter-senses, contradictions, and skepticism. Psychologists (i.e. those who promote psychologism) tried their best to provide an objective account for logic and truth, but at the very end, it was all a failure.

Again, Husserl sympathized with that position, because he came from there. He adored his teacher Brentano, but not to the point of sacrificing what he knew had to be true: that logic cannot be reduced to norms of mental operations. As always, in philosophy, many thinkers are misled by their own prejudices.

Psychologism’s Three Prejudices

Perhaps one of the key aspects of Husserl’s criticisms to psychologisms of all sorts has to do with three basic prejudices that permeate them, plague them, and blind these thinkers, and engage them in futile quests. This is the part where Husserl, for all practical purposes, places the nails of psychologism’s coffin.

First Prejudice

According to Husserl, psychologism’s first prejudice can be formulated this way:

The norms and principles which regulate the mind are founded in psychology. Therefore, it is also evident that normative laws of knowledge must be founded on a psychology of knowledge.

This is the core of all psychologism: reducing all knowledge to psychological operations. If logical laws establish the norms of knowledge, then they are norms of psychological operations.

As Husserl already pointed out there are two sides of logic:

  • Theoretical aspect of logic: which tells us what is.
  • Normative aspect of logic: which tells us what we ought to do.

To understand this distinction, he gives us a good example. Let’s say that someone says something like this:

A good soldier is courageous.

For Husserl, this is a theoretical statement. Don’t be misled by the term "theoretical". It doesn’t mean that it is a conjecture or it is a mere speculation. In Husserl’s sense of the word, this statement tell us what is universally true. In the same sense, the principle of no-contradiction or Modus Barbara, are all theoretical rules of logic.

However, if someone says something like this:

A soldier ought to be courageous.

then, this is a normative statement, because it does not tell us what is, it only tells us what soldiers ought to be.

Now, the question is: which is more fundamental … the theoretical statement or the normative statement? The answer is: the theoretical. A theoretical logical statement tells us what is true no matter what! The normative statement is based on the theoretical statement. The unquestionable self-evident truth that a good soldier is courageous serves as the foundation of the norm that all soldiers ough to be courageous.

Due to how close pure logic and pure mathematics are, Husserl gives us another example to distinguish the theoretical and the normative sides. Let’s take, for example, this formula:

(a + b) (ab) = a² – b²

When we look at this formula, there is no statement about what we ough to think, just what is. This is itself a theoretical mathematical statement. There is no norm established here (no word "ough" anywhere), nor does it describe any psychological process. However, if we said something like this:

To find the product of the sum and the difference between any two numbers, you ought to establish the difference of their squares.

This is a normative statement, and this norm is based on the theoretical mathematical truth.

By the way, people who wish to Neo-Kantianize Frege have said that for Frege logic is a normative discipline. This is false. Although he doesn’t use the term "theoretical", he fully agrees with Husserl in this very important point. There are many statements which show this very clearly, but one passage from "Thought: a Logical Inquiry" will suffice:

Just as "beautiful" points the way for aesthetics and "good" for ethics, so do words like "true" for logic. … To discover truths is the task of all sciences; it falls to logic to discern the laws of truth. The word "law" is used in two senses. When we speak of moral or civil laws we mean prescriptions, which ought to be obeyed but with which actual occurrences are not always in conformity. … From the laws of truth there follow prescriptions about asserting, thinking, judging, inferring. And we may very well speak of laws of [psychological thinking] in this way too. But there is at once a danger here of confusing different things. People may very well interpret the expression "law of [psychological thinking" with "law of nature" and then have in mind general features of thinking as a mental occurrence. A law of [psychological thinking] in this sense would be a psychological law. And so they might come to believe that logic deals with the mental process of thinking and with the psychological laws in accordance with which this takes place. Error and supertition have causes just as much as correct cognition. Whether what you take for true is false or true, your so taking comes about in accordance with psychological laws. A derivation from these laws, an explanation of a mental process that ends in taking something to be true, can never take the place of proving what is taken to be true. … In order to avoid any misunderstanding and prevent the blurring of the boundary between psychology and logic, I assign to logic the task of discovering the laws of truth, not the laws of taking things to be true or of thinking (Beaney, 1997, 325-326).

What is Husserl and Frege’s point here? Very simple, the norms which we should follow if we wish to find the truth are not founded on psychological laws, but rather in theoretical logical laws. These theoretical laws themselves tell us nothing about mental processes or physical or biological laws which operate the brain or the mind. They only express those formal logical relations to express truth.

Second Prejudice

Psychologism’s second prejudice can be formulated in this way:

Logic is about judgments, reasonings, proofs, probabilities, necessities, possibilities, foundations, consequences and other related cocncepts. But judging, reasoning, finding necessities and probabilities, and the like are all psychological processes. Therefore, logic belongs in psychology.

The problem with this argument is twofold. First, it confuses psychological acts with the actual validity content of these acts. One thing is all the psychological operations that lead me to formulate the sentence: "JFK was killed in 1963". It is quite another different thing to say that the truth expressed in this sentence has a psychological basis. In reality, the truth will not depend at all on psychological acts. It will depend only on two factors:

  • Meaning: What the sentence means (its proposition or judgment).
  • Referent: If the sentence is fulfilled in a state-of-affairs (fact)

The proposition expressed in "JFK was killed in 1963" is true and will always be true, even if everyone in the future would think it false. In this sense, psychological processes have little to do with the truths which minds are able to grasp. Truths are independent of our minds.

The second problem stems from the fact that logic and mathematics are sister disciplines, logic has mathematics as its necessary ontological correlate, as we have explained before (see this blog post). This means that if logical truths depend inherently on psychological processes, then that means that mathematics does too. Psychologism’s prejudice regarding mathematics is very similar to logic’s: mathematics is about numbers, yet, we need a psychological acts of "counting" to have numbers, or grouping things together, and so on.

Husserl argues that the number five is not the act of counting to five, nor any psychological representation of the number. A number itself is given in a formal structure in a state-of-affairs, but the act of grasping it is altogether different from the number itself. The same goes too for all the laws and principles of arithmetic, geometry and any other mathematical field. These laws are not themselves psychological acts, but truths-in-themselves which we are able to grasp through a psychological act or process. The proposition "2 x 2 = 4" is true and will be forever true regardless of whether tomorrow we were to believe psychologically that "2 x 2 = 5".

From these facts we have to make several distinctions:

  1. If what we said above is true, then logic and mathematics are ideal sciences, while sciences about matters-of-facts (natural sciences, psychology, anthropology, etc.) are real sciences. The former are a priori, which means that their truths can only be known independently of experience and through reason alone. These would be the realm of relations-of-ideas. On the other hand, real sciences are empirical or a posteriori (based on experience).
  2. In all knowledge, especially in every sience, we have to distinguish three sorts of interconnexions: first, interconnexions of psychological representations, acts of judging, psychological assumptions, and so on, which occur in the minds of scientists; second, the interconnexions of the objects or objectualities being studied by that science; third, the ideal logical interconnexions among the concepts and truths expressed in scientific theories proposed by such science.

As long as we keep these distinctions in mind, we won’t have any problems and confusions regarding what belongs to logic and mathematics as ideal disciplines, and what belongs to other sciences such as psychology.

Third Prejudice

Psychologism’s third prejudice goes as follows:

If we find a logical or matheatical proposition to be true it is because we find it evident that it is true. Evidence is itself a psychological experience or feeling which is somehow psychologically "attached" to the proposition itself. So the truth or falsity of a proposition necessarily depends on this feeling.

Husserl says that this prejudice confuses once again psychological processes with truth using the notion of "evidence". No logical or mathematical principle or law says anything about the feeling of "evidence" we should all experience. Modus Barbara itself is not equivalent to the "feeling of evidence that Modus Barbara is true", since this logical law says nothing about that.

… There is another confusion besides this, though.

The notion of evidence is not itself a feeling that comes out of the blue. It is more an intellectual assertion that what is being given is true. Evidence is founded on the theoretical side of logic. We find the principle of no-contradiction as evidently true for one reason only: because it is true … period. In mathematics the same. It is impossible for us to psychologically represent in terms of imagery all of the numbers in the decimal form of the number "pi", which is itself an irrational number. Despite this, we know that it is evidently true that there is a trillionth interger of the number pi, yet we are not able to represent it psychologically, nor do we know anything about it (psychologically or otherwise). This is because it is an ideal truth (which escapes all psychological representations) that there is a trillionth interger of the number pi. …. We don’t know what it is, but ideally it is there, it exists. We know this a priori.

Husserl’s Conclusion

If all of these prejudices are wrong, and logic (nor mathematics) can be reduced to psychology, we are forced to establish a distinction between two very different realms:

  • Ideal: By "ideal" Husserl means that is existent, but is abstract and independent of the human mind. This is the realm of meanings, essences, true logical relations, numbers, sets, and other categorial forms. This is the realm whose characteristic is that it is atemporal, these truths do not change or are affected by temporal events, in a very real sense they are eternal.
  • Real: By "real" Husserl does not mean "existent". Ideal entities, meanings and relations do exist independently of us. What Husserl means by real [or "reell"] is that it is concerned with particular moments in time. This is the realm of physical objects, which persist and change in time, and also of psychological acts which also occur in time.

Basically, psychologism in all of its forms made the mistake of reducing the ideal to the real, something which generates all sorts of contradictions, counter-senses, fallacies, and so on. Only by supposing the independent existence of an abstract ideal realm, we solve philosophically all of these confusions. Platonism is the way to go. 😀


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

Rosado Haddock, G. E. (2000, October). The Structure of Husserl’s Prolegomena. Manuscrito, 23(2), 61-99.

A Journey to Platonism with Edmund Husserl — 11

On April 28, 2011, in Philosophy, by prosario2000

Hume’s Big Skepticism

Hume was a psychologist, in the sense that he reduced all knowledge to psychological operations. And unlike many of those who came after him, he was very careful in not doubting humanities’ mental faculties. He established a distinction between relations-of-ideas and matters-of-fact. We have discussed that before. He never doubted relations-of-ideas: all circles are round, no matter what. Yet, what about matters-of-fact?

It was here, and not in relations-of-ideas, that Hume’s skepticism was more manifest. A matter-of-fact is, by definition, that sort of truth which is contingent, not logically necessary. It could be otherwise. It is a matter-of-fact that I was born in San Juan, Puerto Rico. But I assure you, if my mom would have taken a trip to Ponce, then it is a possible scenario that I would have been born in Ponce, Puerto Rico … or maybe in New York City … or maybe in Peking … who knows! There are infinite possibilities on how, when, or where I would have been born. There is no logical necessity for me to have been born in San Juan! It is a fact, though.

From the point of view of knowledge, the question is, what are we really given in experience? For Hume, all that we are given from the "outside" world are what he calls impressions: what we see, smell, taste, hear, and touch. For Hume, we are not given actual objects, just impressions. One possible explanation for our impressions is that there are actual objects out very much like the way we sense them. This itself is a matter-of-fact. I remind you that, as a matter-of-fact, this is a just one possibility, and not the only one. There can be vast infinite possibilities of explanations for the impressions we have.

Are these objects "substances" in the Aristotelian sense of the word (an object whose existence is independent of every other object)? Hume was not a practical skeptic … in the sense that if he is standing in the middle of the street, and sees a car approaching him, he will be wise enough to get out of the way. Yet, his problem is one called de jure …. "with what sort of rational right do I derive or infer that there are actually objects (substances) out there when all that we are being given are impressions and nothing more?"

In the same spirit, he also questioned the cause-effect relationship. No one can actually have impressions of something called "cause" (in pure abstraction) and something called "effect" (in pure abstraction). He also goes as far as to debunk Descartes’ notion of the "ego" as the absolute "must exist" for our mental operations to take place. Since the ego is never given in our impressions, and our ego belongs to the realm of matters-of-fact, we cannot actually state its existence.

In other words, as far as it goes with matters-of-fact, we are led to absolute skepticism (theoretical at least). For Hume, there seemed no way out of this. Husserl also learned this from David Hume. And even when Kant tried to circumvent this trying to state that the human mind applies some forms of intuition and pure concepts of understanding, there are several problems with his arguments: first, he is focused solely on human understanding, without taking into consideration any other rational being whatsoever; second, his "pure concepts of understanding" (aka categories) are in reality a mix of Aristotle’s categories along with some conceptualized version of Isaac Newton’s three laws of motion. Husserl took note of that when he called Kantian categories as "mythic" (Logical Investigations, Prol. § 58). Even when Kant wanted to escape skepticism, he could not get away from it fully, hence falling into a sort of relativism, which Husserl would call "specific relativism".

Skeptic Relativism

Psychologism leads to relativism … regardless how much they try to guarantee certain knowledge from their psychologistic prejudices. That is the lesson Husserl taught in the "Prolegomena of Pure Logic" … and still teaches us.

Still … Husserl’s rejection of psychologism as relativism is more refined than Frege’s. Although Frege is right that we should reject psychologists reduction of all logic and mathematics to subjective representations, as we have said before, he had the distinct quality of distorting his opponents’ views, practically reducing all of their doctrines to some gross relativism that many of them never held.

Again, Husserl’s criticism is more effective, because he was much fairer to his opponents. First, he made a distinction between relativists and those who didn’t proclaim themselves as absolute relativists. Second, he made a distinction between the intended purpose of several psychologists, and their unintended logical outcomes. So, even when a particular psychologistic philosophical opinion (or a similar one) would not proclaim itself to be relativistic, Husserl would reveal through logical deduction that their suppositions derive nothing more than relativism, even when the holders of these doctrines don’t intend to.

Husserl recognized that there were two sorts of relativism: individual relativism, and specific relativism.

Individual Relativism

Individual relativism is the form of gross relativism Frege has in mind when accusing all psychologists for being relativists. Yet, as Husserl points out, this form of relativism is so absurd that we should doubt if anyone has taken it seriously. It practically reduces all truth to subjective opinion. In other words, the famous: "What is true for me may not be true for you."

Individual relativism is the opinion that there is no objective truth. As every philosopher knows, this statement is self-defeating. Saying that "There is no objective truth" is equivalent to saying that "It is objectively true that there is no objective truth." There is no truth-in-itself (to use Bolzano’s famous phrase) different or distinct from my mental operations. By establishing all logical principles (such as the principle of no-contradiction) to mere subjective opinions, there cannot be any sort of philosophy built on this. And as Barbara Ehrenreich would say: this sort of relativism, if practiced, cannot even be the basis for any sort of normal conversation.

Specific Relativism

This is a milder form of relativism, which can be called anthropologism. It basically states that any statement is true in relation to a species (hence the word "specific"). In this case, many hold that the set of propositions we call "truth" is in reality truth in relation to humans, because our mental or biological constitution can be different.

For Husserl, Kant tried to avoid individual relativism and the sort of skepticism he so rejected of Hume by adopting an anthropological position: knowledge is "guaranteed" given that humans have such and such faculties.

Let’s see how Husserl rejects and refutes various sorts of anthropologisms:

  • Specific relativism says this: each species is capable of judging that it is true what, according to their constitution or their own psychological processes, should be held as true.

    Husserl says that there are two ways this anthropological assertion is wrong. First, any statement (proposition) is either true or false if it is fulfilled or not in an actual state-of-affairs. This means that even when other species hold some true statement to be false, then they are holding as true a false belief. In this aspect, both Frege and Husserl are in perfect agreement, one thing is being true, and another thing is holding or believing something as true. Regardless of any species’ constitution, if a species believes that something is true or false, does not mean that it is true or false.

    It is a contradiction (or in Husserl’s words "counter-sense") to be talking about a "truth for someone" or a "truth for one species". In his own words: truth is one and identical to any species of rational beings able to grasp it, even if it is for humans, for angels, or for the gods.

  • A specific relativist would say: It could be that the words "true" and "false" mean a different thing to another species, since maybe their own mental constitution wouldn’t let them grasp the logical laws which are implied in our own concept of truth: such as the principle of no-contradiction or the principle of the excluded middle.

    Husserl responds by saying that if another species uses the words "true" and "false" to mean different things than what we mean, then it is a problem of the meaning of the words "true" and "false". Let us never to confuse name and meaning (as Frege and Husserl pointed out in their respective works — see here for more details here and here). When we are concerned about truth, we are really not concerned about words themselves in their quality as signs, but on what the words mean (i.e. propositions), and their fulfillment in a state-of-affairs (or "facts").

    It can be possible that there are extraterrestrial beings who are not able to grasp logical laws as the principle of no-contradiction. If their use of the word "truth" is the same as ours, and still, they wish to negate this particular logical principle, then their negation would be false, even if they thought that such a possibility might be true. However, if their word "truth" means something else altogether, then it is inherently a problem of meaning: they would not be grasping any truth in our meaning of the word. In such case our meaning of the word "truth" is completely unaffected, as are logical laws themselves.

  • The specific relativist might say: The constitution of a species is a matter-of-fact, and only matters-of-fact can be derived from other matters-of-fact. The concept of truth and logical laws are matters-of-fact because they are founded on a species’ existence which is itself a matter-of-fact.

    Husserl’s response that this is a counter-sense once again. A matter-of-fact is a singular event (a sunset, a star in the sky, a Pres. George Bush who doesn’t know how to spell "nuclear", etc.) In other words, they are temporal events. Yet truths themselves are not subject to the cause and effect relations in time. This confusion happens because psychologism mixes the psychological act of constituting a truth, and the truth-in-itself. Of course we carry out all sorts of mental operations to grasp the truth that "2×2=4". There is absolutely no issue about this. However, the truth "2×2=4" does not depend on us. We can constitute a truth, but we do not create it. Therefore a universal truth is never founded on matters-of-fact, a proposition is only true if it is fulfilled by a matter-of-fact, not that a matter-of-fact "derives" or "infers" a universal truth.

  • The specific relativist might argue: If all truth has an exclusive basis on the constitution on the human species, then if there were no human species, then there would not be any truths at all.

    This would fall into the same problem as in the case of individual relativism, it is self-defeating because it establishes as objective truth that there are no objective truths at all.

  • The specific relativist might argue: It can be possible that given a certain specific constitution, such a constitution would lead a species mentality to conclude as truth that there is no such constitution.

    This is another counter-sense. For Husserl, truth is nothing more than a network of true propositions which are necessarily correlated to reality (a whole network of states-of-affairs). What are we to say about this sort of anthropological argument, then? That there is no reality, or that it doesn’t exist except to humans? And, what would happen if all humans disappeared, is reality going to disappear along with it? Definitely, we are moving in contradictions.

    It can be possible for a species to have a constitution which can lead it to a false claim. However, it is quite another thing altogether to claim that it would be true to claim that there is no specific constitution because it is itself based on an existent constitution.

    By the way, it would not be less absurd if an anthropologist claimed that if such a species recognized the truth of its own specific constitution, then this truth would be founded in such constitution. If they claim that truth is dependent on the species Homo (us!), then this dependence can only be understood causally and according to the laws which rule such causal relation in the constitution. Husserl says that in this case, we would have to claim that the truth "this constitution and these laws exist" would be explained by the fact that they temselves exist, which means that they would be founded on themselves. At the same time, the principles which would agree with such explanation would be identical to these laws themselves. This is non-sense: the constitution would be its own cause, founding itself on laws that would cause themselves by founding themselves on themselves, etc.

  • Husserl points out that one further consequence of any argument presented in favor of a relativity of truth (and anthropologism is no exception), is that it implies the relativity of the universe itself. The universe is nothing more than the objectual unity of all states-of-affairs, which are necessarily correlated by all truths about these states-of-affairs. We cannot relativize truth, and at the same time state that there is a universe independent of our own constitution. If there is a truth for a species, then there is a universe for a species. So, if the species disappears, would the universe disappear?

    This would be obvious to everyone, but if we reflect a little bit about it, we become aware that our own ego and its psychological acts belong to this universe, which would also mean that every time I say "I exist" or "I have such and such experience", it would be instantly false in a truth-relativistic point of view.

    So, question: if our constitution changes, would the universe change along with us to fit our own constitution? And would our constitution, which is part of the universe, change if the universe changes? Nice circle, isn’t it?!

Relativism in General

Husserl, reminding us of Hume, says that all matters-of-fact are contingent: they could be otherwise. If logic is founded on matters-of-fact, then its laws would be contingent, yet they aren’t. They are the foundations for all sciences, and there is a reason for that: any science which rejects these logical laws would be inherently and necessarily self-contradictory. It nullifies itself. We cannot derive any universal logical rule or law from causal and temporal matter-of-fact. Any effort to do so would be self-defeating.

Since psychologism in all of its forms (even in the case of anthropologism) require that logical laws be matters-of-fact, they open themselves to the idea that logical laws are contingent, and there would be absolutely no reason to object any contradictory theory. Remember what Hume taught us: we can be absolute skeptics regarding matters-of-fact, not about relations-of-ideas. The problem with psychologism in Husselr’s time is that it opens the door to being skeptical about absolutely everything, including relations-of-ideas.


Hume, D. (1975). Enquiries concerning human understanding and concerning the principles of morals. L. A. Selby-Bigge & P. H. Nidditch (eds.). Oxford: Clarendon Press. (Original work published in 1777).

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

Rosado Haddock, G. E. (2000, October). The Structure of Husserl’s Prolegomena. Manuscrito, 23 (2), 61-99.

A Journey to Platonism with Edmund Husserl — 10

On April 23, 2011, in Philosophy, by prosario2000

Those who know Husserl very well may ask why did I spend a good deal of time explaining the journey to Platonism describing first his logical and mathematical philosophy, and not with Husserl’s critique of psychologism? If you look at Husserl’s Logical Investigations it is the other way around. Well … that may be so in the structure of Logical Investigations, but in chronological terms, chapter 11 of the "Prolegomena of Pure Logic" was written first, the whole critique of psychologism was written later.

Someone said once (I fail to remember who) that Husserl’s "Prolegomena" represented the most formidable refutation of psychologism ever. I agree. Not even Frege was this good. Frege’s arguments against psychologism and other antiplatonist arguments in general are good, but he has a fatal flaw. If you have been reading these series, especially regarding his review on Husserl, you realize that Frege had this tendency of exaggerating or even distorting his opponents’ opinions. This is unfortunate. Contrary to what people believe, distorting an opponent’s opinion only weakens your own position.

On the other hand, Husserl was extremely fair, and there are many reasons for it. First, he did come from that tradition, so he knows all psychologistic positions, all accross the spectrum. He spends a great deal of time making all sorts of distinctions between psychologists who were more extreme, and those who were not. Second, he usually was very honest about his research and thoroughness. Third, because he was very hard on his errors of the past, errors which went through subtle changes from more extreme to more moderate. And fourth, because he was also critical of philosophers whose positions were close to his own, without distorting their opinions.

What is Logic?

It is difficult to know exactly what logic was in the nineteenth century. So many people held so many positions at that time, that it confused philosophers of every tendency. Psychologism was "in!" at that moment, because, since the time of John Locke, everyone believed that the principles of knowledge could only be achieved by examining our subjective mind. Immanuel Kant tried to overcome the problem by stating that the human mind had faculties and concepts which guarantee knowledge, because we all share these faculties.

Logic, in this sense, was reduced to what people always thought since the beginning: "it is the art of correct thinking." It posits all sorts of rules for us to follow if we want to carry out a thinking process that will lead us to the truth, hence to knowledge. Therefore there are two things which might be said about logic from a psychological point of view:

  • Logic is a technique: an instrument which benefits our thinking processes.
  • Logic is normative in nature: which means that it establishes the "rules for right thinking"

Husserl will beg to differ on both accounts, and in the "Prolegomena of Pure Logic" he tells us why logic is theoretical: and by this term he does not mean that logic is speculation, it means that logic does not tell us how we ough to think, but tell us formally what is.

When Sciences Go Bezerk

One of the big problems psychologism has is that it wants to submit all forms of knowledge, even formal knowledge, to psychological thinking. If logic is the "art of right or correct thinking" (and notice the word "thinking"), then logic is nothing more than a branch of psychology. In the "Prolegomena", Husserl complains against antipsychologists, because they pretended to beat psychologism while they were conceiving logic as "the art of correct thinking". Hence, when they debated psychologists, psychologism kicked their behind every single time.

Husserl states that antipsychologists are essentially correct, but the reasoning with which they pretend to say that logic does not belong to psychology is seriously flawed by the supposition that logic is a set of rules for us to think.

So, the question is the following. Does logic belong to psychology or not?

In here, Husserl says that we can look at all sciences around us, and see that some are general, and some are more specialized … but nothing too specialized. For instance, we know that there is a science called zoology, yet we don’t see the GRAND field of "science of lions" (or "lionology") or the "science of chairs" ("chairology") anywhere. At least not as a field! Of course, a particular scientist may dedicate his or her whole life to lions and chairs, but it still doesn’t merit a specialized field for all fans of lions and chairs out there!

But what happens when a particular field of science is too broad? What happens if a field occupies issues of another field? What would happen if zoology would incorporate something like botany? Everyone would agree that there is what Husserl would call, in Greek, a "μετάβασις εἰς ἄλλο γένος" (Isn’t Greek pretty? It is pronounced "metábasis eis allo génos") or a "trangression to another genus (field)". Botany is about plants, not animals … therefore it should never be considered a branch of zoology (the science of animals). In this case, botany is a field in its own right.

This is exactly what Husserl thinks about turning logic into a branch of psychology (which is what psychologism is). Psychology is an empirical science, hence, it deals with matters-of-fact. On the other hand, logic is its own field, because it belongs to the realm of relations-of-ideas (or truths-of-reason). So, psychologism would be, for all practical purposes a "μετάβασις εἰς ἄλλο γένος". Psychologism is trying to present as united two fields which are not.

Empirical Consequences of Psychologism

Among psychologists we can count on John Stuart Mill as one of its greatest representatives. Despite the fact that he was considered one of the greatest minds of his times, Frege could not resist the temptation of making fun of him, especially with Mill’s assertion that mathematics is somehow abstracted from sensible experience. Of course, I cannot resist the temptation of sharing with you how Frege made fun of him. This is one of my favorite passages in The Foundations of Arithmetic.

John Stuart Mill … seems to mean to base the science, like Leibniz, on definitions, since he defines the individual numbers in the same way as Leibniz; but this spark of sound sense is no sooner lit than extinguished, thanks to his preconception that all knowledge is empirical. he informs us in fact, that these definitions are not definitions in the logical sense; not only do they fix the meaning of a term, but they also assert along with it an observed matter-of-fact. But what in the world can be the observed fact, or the physical fact (to use another of Mill’s expressions), which is asserted in the definition of the number 777864? Of all the whole wealth of physical facts in his apocalypse, Mill names for us only a solitary one, the one which he holds is asserted in the definition of the number 3. It consists, according to him, in this, that collections of objects exist, which while they impress the senses thus, ⁰0⁰, may be separated into two parts, thus, 00 0. What mercy, then, that not everything in the world is nailed down; for if it were, we should not be able to bring off this separation, and 2 + 1 would not be 3! What a pity that Mill did not also illustrate the physical facts underlying the numbers 0 and 1! (p. 9)

Here is another passage:

[For Mill] it appears that his inductive truth is meant to do the work on Leibniz’s axiom that "If equals are substituted for equals, the equality remains." But in order to be able to call arithmetical truths laws of nature, Mill attributes them a sense which they do not bear. For example, he holds that the identity 1=1 could be false, on the ground that one pound of weight does not alwayss weigh precisely the same as another. But the proposition 1=1 is not intended in the least to state that it does (p. 13).

Although with much less fun, but still remaining highly critical, Husserl sees this same pattern in John Stuart Mill’s work regarding logic. For example, one point of interest of any philosopher of logic is the principle of no-contradiction. This principle states that a proposition and its negation cannot both be true in the same sense at the same time. In symbolic logic we represent it this way:

~ (A & ~A)

Where "A" is any proposition whatsoever ("There is a cat on the roof", "Obama is United States’ president", "The Joker is Batman’s foe"), "~" is the symbol for negation ("no", "not", "it is not the case") and "&" is a conjunction ("and"). In other words, this formula is read like this: "It is not the case that A and not-A". Because Mill is so darn stubborn insisting that all knowledge is abstraction from facts, Husserl criticizes Mill for saying that the principle of no-contradiction is derived from experience.

John Stuart Mill, it is well known, held the principle of [no] contradiction to be ‘one of our earliest and most familiar generalizations from experience’. Its original foundation is taken by Mill to be the fact ‘that belief and disbelief are two different mental states’ which exclude one another. This we know — we follow him verbatim — by the simplest observation of our minds. And if we carry our observation outwards, we find that here too light and darkness, sound and silence, equality and inequality, precedence and subsequence, succession and simultaneity, any positive phenomenon, in short, and its negation, are distinct phenomena, in a relation of extreme contrariety, and that one of them is always absent when the other is present. ‘I consider the axiom in question’, he remarks, ‘to be a generalization from all these facts.’

Where the fundamental principles of his empiricistic prejudices are at stake, all the gods seem to abandon Mill’s otherwise keen intelligence. Only one thing is hard to understand: how such a doctrine could have seemed persuasive. It is obviously false to say that the principle that two contradictory propositions cannot both be true, and in this sense exclude one another, is a generalization from the ‘facts’ cited, that light and darkness, sound and silence, etc., exclude one another, since these are not contradictory propositions at all. It is quite unintelligible how Mill thinks he can connect these supposed facts of experience with the logical law. (Prol. § 25).

Husserl is right: belief (defined as a mental state) is not a proposition, sound and silence are not propositions, light and dark are not propositions, and so on. But here is Husserl’s point: how can John Stuart Mill derive an absolute, necessary, universal logical proposition, from non-absolute, contingent, and singular experiences? What process leads us from one to the other? How can a logical law be a generalized statement from our physical experience in this world? Mill never says how this is so. This is precisely what David Hume criticized about induction.

And this is one of the basic problems with psychologism all accross their spectrum. Even David Hume, an rabid empiricist and skeptic, was far more careful than this!

First Consequence of the Empirical Supposition in Psychologism

Husserl says that psychologists want to legitimize the validity of logical principles as universal and necessary (at least for us), but from a psychological point of view: looking at logical laws as generalizations from sensible experience.

Here is the first reason why it won’t work: From vague foundations you can only derive vague principles ("vague" as opposed to "exact"). The problem with psychology as an empirical science is that its laws can only be probable, never absolutely exact as logical laws are. Since logic is necessarily correlated with mathematics, then also mathematics, which consists of a whole set of exact principles, rules, and laws, would automatically be considered a branch of psychology. So, psychologism is never able to account how it is possible that from the vague laws of psychology we can derive the exact laws of logic and mathematics.

Second Consequence of the Empirical Supposition in Psychologism

Another problem that we have is that psychology is an empirical science, therefore, all of its laws are known by contrasting them with experience. This is not the case in logic, whose rules are known a priori (this means that these rules are known through reason alone, with no reference at all to experience).

For Husserl, the combination of these two consequences generate other unintended consequences. Supposing that all of logic as somehow psychological would mean that no statement can be taken to be absolutely true, but a vague and probable generalization of experience. One of the things Husserl learned from Hume, is that induction cannot guarantee absolute knowledge, only probable ones, because no one can tell you with absolute certainty that similar events in the future will resemble the past. It can always be open to other outcomes. As a result all propositions become probable if their validity relies in operations of the human mind. But think what this would imply: a non-knowledge! Exactly the opposite of what psychologists are searching. Take this proposition, which, by definition, would be only probable (never absolute):

All knowledge is only probable.

Let’s establish this a (let’s say) 90% of probability. Fair enough! Now, through a process of iteration, I can say:

The statement ‘All knowledge is probable’ is probable.

The statement ‘The statement "All knowledge is probable" is probable’ is also probable.

The statement ‘The statement "The statement "’All knowledge is probable"’ is probable" is also probable’ is also probable.




And we could continue ad infinitum, endlessly, each with its own probability. When that happens, the probability of the original proposition being true converges to 0%. In other words, unintentionally, psychologism by its own theory, denies any knowledge whatsoever. (I know that Husserl must have had fun when he wrote this critique).

And even if psychologists wanted to make logical laws as natural laws, we have to ask, how is this statement justified at all at any level of psychologistic literature? The reason for this confusion is that many psychologists actually confuse the causal laws of nature with the non-causal logical laws, even though they try the best to derive one from the other.

Even if they want to define logic as the art of correct thinking, and define "correct" thinking as the way people "normally" think, their feet are too deep in the mud. How many people don’t have so many misconceptions of reality that they actually believe in contradictory things? And how do you place a probable value to that? What guarantees you that the exception, not the rule, are the ones thinking straight?

Third Consequence of the Empirical Supposition in Psychologism

The third consequence of psychologism is that it would interpret logic in terms that are really strange and foreign to it. If logic told us normative principles of thinking, they would have at least some psychological content: some reference to thought processes. Yet, we can find absolutely no trace of matter-of-fact, sensible experience, or thought processes anywhere in logical laws.

Normative statements say: "… you ough to …" Now let’s take Modus Barbara, a well known logical rule.

If all As are Bs
If all Bs are Cs
Then: All As are Cs

So far so good! Well … may I ask you, my dear reader, where is the "ough" part of this logical rule? The word "ough" is nowhere to be found! For Husserl, this is a theoretical rule … it tells us what is, not how we ough to think. Now if I said something like: "If it is a fact for you that all As are Bs, and that all Bs are Cs, then you ough to think that all As are Cs." … then this statement is normative and does tell us how to think. This statement does have psychological content.

If we want to make logical laws be empirical in some way, then we should look at what empirical or natural laws really are. Empirical laws (conceived in a nomological-deductive manner) or natural laws, along with certain circumstances, do explain phenomena. Therefore, they all have empirical content. Even the most abstract physical laws, which seem to resemble in so many aspects logical and mathematical laws, cannot justify themselves without some reference to experience.

For Husserl, from the point of view of knowledge, it is clear that the laws of natural science which refer to facts, are are fictions with fundamento in re (founded on the thing), in other words, founded on the objects of experience. They are, in Popperian terms, conjectures which have to be tested in experience. … Interesting! These physical theories are just a very small set of an infinite horizon of possible theories which may fit experience. We choose the ones we have because they are the simplest ones which can explain all the phenomena we witness.

Yet, none of this content can be found in any logical law, nor in mathematics. There is no psychological (nor any other empirical) matter-of-fact in a statement like "3 > 2": it doesn’t talk about psychological processes, nor oranges, nor computers. Psychologism is off the mark in this one.

For Husserl, it is undeniable that our knowledge of logic and mathematics are the results of mental processes … but be careful! The fact that there are psychological processes to know "3 > 2" does not mean, that logical and mathematical statements in some way refer to psychological matters-of-facts. Psychologists make this confusion constantly. One thing is the psychological activity of counting to "3", and another the 3 itself.

Combine all of these three consequences of psychologism, and what do you have in the end? Something very simple. If psychologism is true, then no knowledge is possible… and if this is true, then we are led to skepticism. That will be the subject of our next blog post.


Frege, G. (1999). The Foundations of Arithmetic. Evanston: Northwestern University Press.

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

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A Journey to Platonism with Edmund Husserl — 9

On April 21, 2011, in Philosophy, by prosario2000

The "Duhem-Quine Thesis" … A Misnomer in Philosophy if I Ever Heard One

Some people have asked me, after knowing that I am a platonist, what am I to make of the "Duhem-Quine Thesis". When that happens, I point out to them that the "Duhem-Quine Thesis" has a lot in common with unicorns in a very important aspect: there is no such thing!

Pierre Duhem said one thing, and W. V. O. Quine said a very different thing. Let’s start with Quine. For Quine the whole of knowledge is precisely that, a whole unit, a whole network of propositions which interdepend on one another, always subject to revision in light of recalcitrant experience. Yet, despite the fact that many people rushed to embrace this proposal in order to reject the analytic and synthetic distinction, other people have some problems with this. To summarize Quine’s proposal: nothing is sacred, everything is subject to revision in light of recalcitrant experience.

Pierre Duhem
Pierre Duhem (1861-1916)

Duhem was far more careful than that, as philosopher of science Donald Gillies has pointed out. Pierre Duhem did recognize that in physics … and only in physics … there seemed to be some sort of network of propositions which interpret particular phenomena. Let’s say, for instance, that I wish to throw a rock at a certain angle upwards so that it lands some 10 feet away from me. I can predict which amount of force will be necessary and the energy required for that rock to land 10 feet from me. …

And that’s the trick … isn’t it? To make that "sole" hypothesis, I have to suppose a whole baggage of Newtonian theory: theory of mass, of force, of energy, of how are these concepts related to something like velocity and speed, or acceleration, etc. Then I’ll have to include concepts like gravity, gravitational constant, the relationship between gravitational acceleration and masses, etc.

So if you carry out an experiment, you are not just testing one little teeny weeny hypothesis. Essentially you are testing a whole theoretical group of scientific suppositions and statements which interpret these phenomena, and tell you how to run your experiment. As Duhem said: "An Experiment in Physics Is Not Simply the Observation of a Phenomenon; It is, Besides, the Theoretical Interpretation of This Phenomenon" (Duhem 1905/1991, p. 144).

Don’t believe me? Here, let Duhem explain it to you:

Go into this laboratory; draw near this table crowded with so much apparatus: an electric bettery, copper wire wrapped in silk vessels filled with mercury, coils, a small iron bar carrying in mirror. An observer plunges the metallic stem of a rod, mounted with rubber, into small holes; the iron oscillates and, by means of the mirror tied to it, sends a beam of light over to a celluloid ruler and the observer follows the movement of the light beam on it. There, no doubt, you have an experiment; by means of the vibration of this spot of light, this physicist minutely observes the oscillations of the piece of iron. Ask him now what he is doing. is he going to answer: "I am studying the oscillations of the piece of iron carrying this mirror?" No, he will tell you that he is measuring the electrical resistance of the coil. If you are astonished and ask him what meaning these words have, and what relation they have at the same time perceived, he will reply that your question would require some very long explanations, and he will recommend that you take a course in electricity. (Duhem 1905/1991, p. 145).

So, again, the problem anyone ignorant in physics has is that he or she will never understand what is going on in an experiment, hence will not have the necessary background to interpret it. As Duhem argues very well, if I’m ignorant of the life of the sea, I could not understand: "All hands, tackle the halyard and bowlines everywhere!" Regardless of my own particular understanding of this order, the men on the ship understand it very well and carry out those orders (Duhem 1905/1991, p. 148).

Experiments are only possible, if there is a previous scientific theory to interpret such results (Duhem 1905/1991, pp. 153-158).

But notice that, for Duhem, unlike Quine, he restricts it to physics. This is not applicable to physiology or other fields … and much less to mathematics and logic. I agree with Duhem to a certain extent, but some aspects of physiology have much theoretical baggage behind it too, other aspects of it don’t.

So, when people talk about the "Duhem-Quine Thesis" is in reality Quine’s thesis, not Duhem’s.

Husserl’s Conception of Science

Edmund Husserl was not a philosopher of science, but his philosophy was definitely inspired by physics, a discipline he so admired. In part, his philosophical enteprise, even his phenomenological research, was directed to legitimize science, for its incredible value to society.

Yet, he knew that for science to be reliable it had to obey logical and mathematical laws. How did Husserl think science builds its theories and interprets observations? Here is a direct quote from his Logical Investigations.

"Empirical laws" have, eo ipso, a factual content. Not being true laws, they merely say, roughly speaking, that certain coexistences or successions obtain generally in certain circumstances, or may be expected, with varying probability, in varying circumstances. (Vol. I. Prol. § 23).

What is he saying here? First of all, science operates according to "empirical laws". These laws are not "true laws" in the sense that they are not as universal and necessary as logical and mathematical laws are. They seem to be laws that are at least valid in our own universe, in our own reality. However, following Leibniz, unlike natural laws, Husserl regarded logical and mathematical laws to be valid in every possible world.

These empirical laws, posited by scientific theories, seem to be interconnected logically among themselves. If we could represent it in some way, let’s do it this way:

L1 & L2 & L3 & L4 . . . Ln

According to Husserl, these laws do not operate by themselves, because they have "no factual content". They establish what laws operate in the universe, but they don’t tell us about actual events. Why is that? Because events operate according to these laws and "varying probability, in varying circumstances"! In other words, we could represent Husserl’s views on how science explains phenomena this way: science formulates theories which posit some regularities called "laws" (L), and that these laws along with certain circumstances (C), will lead to an explanation of phenomena (P).

L1 & L2 & L3 & L4 . . . Ln

C1 & C2 & C3 & C4 . . . Cn

Now … if you are versed in philosophy of science, you will be very surprised to see Husserl formulating in a sketchy way Carl G. Hempel’s deductive-nomological scheme. as he proposed it in the 1940s. In a very short passage, Husserl shows how way ahead of his time he was!

The Formal Components of Scientific Theories

Science, like every other field, is a large theoretical group of propositions. As we have seen in Husserl’s theory of sense and referent, scientific propositions, like all propositions, refer to states-of-affairs. Let’s examine propositions for a moment.

Acts of Meaning

The only question we never really answered in these series is how do we formulate propositions? Which acts of consciousness intervene in these process? Remember, our primal constitution is of states-of-affairs. For example, we can constitute a white sheet of paper on the desk. However, by other acts of consciousness, I mentally constitute another different sort of form to refer to that state-of-affairs. That mental act is what Husserl calls a meaning act, which makes possible for me to say "There is a white sheet of paper on the desk". Just as formal-ontological cateogories, the word "is" does not have any sensible correlate (such as the sheet of paper itself), but it establishes the existence of such a white sheet in a particular manner. In the same way, I can constitute Megan taller than Mary, but by a meaning act I can propose that "Megan is taller than Mary".

And the word "is" in this context does correlate with a categorial form, but not it is not a formal-objectual category … but a meaning category. What meaning categories do is to structure objectualities in such a way that it is possible to communicate what we want to propose in a meaningful manner. Let me give you an example of what I mean.

Imagine someone who would tell you "table Zingale the outside sits porch at". Of course, this is not a meaningful thing to say … in fact it is not a statement at all, since statements are meaningful. Yet, if you follow the rules of grammar, then you can say that "Zingale sits at the table outside the porch" … interesting place to sit.

Husserl says exactly the same things. Meaning categories let us arrange objectualities and actions in a meaningful proposition. For these propositions to be meaningful, this arrangement has to follow universal and necessary grammar rules for meaning. We are not talking here about the rules of grammar in a specific language. Even when in English the verbs are in the middle, and in German at the end, it makes no difference for Husserl. The grammar he is talking about has to do with the way meaningful propositions are arranged in the abstract sense. This is a realm which linguistics knows very well … as is the idea of a Universal Grammar proposed by Noam Chomsky. His point is the same as Husserl’s … underlying every language on Earth there are some basic structures shared to express states-of-affairs. The only difference between Chomsky and Husserl is that the former established it in naturalistic terms, the latter in a priori terms.

Like all propositions, scientific propositions have material meanings (concepts or meanings of proper names which refer to objects) and formal concepts (which refer to categorial forms). Through categorial abstraction, let’s get rid of all of the material concepts and states-of-affairs, and what do you have? The form of the proposition in its purity, or meaning categories, which include, but are not limited to:

  • Subject – Predicate Structure
  • Forms of Plural
  • Conjunction ("and")
  • Disjunction ("or")
  • Implication ("if … then ….)
  • Negation ("no", "not")

Also, if these propositions are associated in a deductive or logical manner (like scientific laws are), we are able to see also these deductive relations among them in their purity, without appealing to any sort of sensible content.

The Relationship between Formal Logic and Mathematics: a Mathesis Universalis

Again, science is made up of propositions, which refer to states-of-affairs. Let’s remember that if a proposition is true, it is because it is fulfilled in a state-of-affairs, or it has a state-of-affairs as its correlate. Yet, if science proposes a set of logically and deductive related propositions, they correlate with a whole network of states-of-affairs.

Formalize propositions and states-of-affairs through categorial abstraction, and you will have meaning categories in their purity deductively and logically interconnected with one another on the side of propositions, and correlated with these are formal-objectual categories in their purity on the side of states-of-affairs.

These meaning categories are the basis of formal logic, while formal-objectual categories are the basis for mathematics. We can see here the relationship between the two … but how do they integrate in a "mathesis universalis"? Here is how Husserl solved the problem. He divided the correlation of formal or pure logic on the one hand and pure mathematics on the other in three different strata.

First Logical and Mathematical Stratum

Like we have seen, logic is made up of meaning categories, forms of plural, conjunction, disjnction, implication, negation, subject-predicate structures, and so on. This is the stratum of a priori universal grammar, where meaning categories arrange objectualities into meaningful propositions. This is called by Husserl a morphology of meanings (in other words, how meaning categories "shape" propositions). This stratum is ruled by a priori laws which he called laws to prevent non-sense.

On the side of mathematics we have formal-objectual categories, which formally "shape" and structure objects in states-of-affairs. Husserl called this a morphology of intuitions or morphology of formal-objectual categories. Here we find formal-objectual categories such as: cardinal numbers, ordinal numbers, sets, relations, parts-whole, and so on.

Second Logical and Mathematical Stratum

On top of this first stratum, we find that propositions can be organized deductively according to simple syllogisms (as Aristotle proposed). For example, take Modus Barbara:

If all animals are mortals
If men are animals
Then all men are mortals

Let’s get rid of the material components, and we will have this simple form of deduction:

If all As are Bs
If all Bs are Cs
Then all As are Cs

In this stratum, truth is not really a concern, only the forms of deduction count, just like the one expressed in this case.

These deductive laws are a priori, and they are called by Husserl laws to prevent counter-sense (contradictions).

In a still upper level (not yet the third), we integrate in this logical stratum the notion of truth and similar concepts, where only true propositions are concerned. He called it the logic of truth.

On the side of mathematics, we find a whole set of disciplines founded on the formal-ontological categories on the first level. For example, with the notion of cardinal number, and other sorts of numbers, we can develop arithmetic as a discipline. On the basis of sets, we can develop set theory. On the basis of part-whole categories, we can develop mereology … and so on. For these disciplines to progress, they use deductive laws of logic in this logical stratum. So, we start to see a gradual integration of mathematics and logic.

Third Logical and Mathematical Stratum

Then there is a third logical level where pure logic becomes a theory of all of the forms of theories or a theory of deductive systems. In this level, a logician is not limited to the simple logical deductions we find in the second stratum, but he or is free to posit and explore exhaustively other formal deductive systems. The only rule of this game is to preserve truth in virtue of their deductive forms.

In fact, that is what logicians today actually do. Husserl had no idea at the time how this was, since, like Frege, he blamed psychologism for this lack of development at that time. Actually, ever since Frege, there was an explosion of search for alternative deductive systems. In this sense, not being a logician himself, Husserl did foresee what logic would become as time went by.

On the side of mathematics, mathematics becomes a theory of manifolds, a mathesis universalis, where a mathematician can posit other mathematical concepts (as in Husserl’s time: negative roots, sets, fractions, and so on) or even add and subtract some mathematical axioms (such as the elimination of the axiom of the parallels, or the creation of rules regarding negative roots or negative numbers, how to handle fractions, etc.). These mathematicians would explore exhaustively all of the consequences of these systems, whose validity will depend on absolute consistency. The logical deductive systems developed at this logical level can be used in this stratum too. For Husserl, the completeness of mathematics should be kept in mind in this stratum. Today, Gödel’s theorems ruined any expectation on the completeness of mathematics, but in a way it can be kept as a sort of Kantian ideal guide for this theory of manifolds to operate fully.

In this way, each logical stratum has as its ontological correlate a mathematical stratum. The correlation is not perfect, but they do explain the relationship between logic and mathematics. Here below is a graphical representation of everything we have just explained.

Husserl's Theory of Logical Strata

Some Interesting Facts …

Rudolf Carnap is known to have made the distinction between formation rules and transformation rules, and this went through history of logic as being his particular contribution to the subject. But we know as a matter of fact that Carnap was pretty much familiar with Husserl’s Logical Investigations, and used Husserlian terminology extensively in both of his first major philosophical works: On Space and The Logical Structure of the World. However, due to his relationship with members of the Vienna Circle who were pretty much anti-Husserl, he wanted to water down Husserl’s contributions to his philosophy, especially in The Logical Structure of the World.

Carnap made this distinction between formation rules and transformation rules in his Logical Syntax of Language, yet it smells that it is one of those occasions he never attributed Husserl the original distinction in Logical Investigations. The laws to prevent non-sense are the Carnapian formation rules, while the laws to prevent counter-sense are the Carnapian transformation rules.

Wadda ya know!

Unfortunately, many scholars who have focused too much on Husserl’s phenomenological doctrine have ignored completely this aspect of Husserl’s work I’ve just presented above. This aspect of Husserl’s philosophy has been worked out by Verena Mayer, Dallas Willard, Claire Ortiz Hill, Guillermo E. Rosado Haddock, Rudolf Bernet, Iso Kern, Eduard Marbach, among other scholars (though very few). For all those interested, the first version of this doctrine appears in Logical Investigations, in chapter 11 of the "Prolegomena to Pure Logic", on his Fourth Investigation, and the Sixth Investigation. For the most detail version of it (the most elaborated one we have to date), we can find it in Husserl’s Formal and Transcendental Logic.


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

Duhem, P. (1991). The aim and structure of physical theory. US: Princeton University Press. (Original work published in 1905).

Gillies, D. (1993). Philosophy of science in the twentieth century: four central themes. Oxford & Cambridge: Blackwell.

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

Husserl, E. (1969). Formal and transcendental logic. (D. Carns, Trans.) The Hague: M. Nijhoff. (Original work published in 1929).

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

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

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

Rosado Haddock, G. E. (2000, October). The Structure of Husserl’s Prolegomena. Manuscrito, 23 (2), 61-99.

Rosado Haddock, G. E. (2006). Husserl’s philosophy of mathematics: its origin and relevance. Husserl Studies, 22, 193-222.

Rosado-Haddock, G. E. (2008). The young Carnap’s unknown master: Husserl’s influence on Der Raum and Der logische Aufbau der Welt. US: Ashgate.

Mayer, V. (1991). Die Konstruktion der Erfahrungs Welt: Carnap und Husserl. In W. Spohn (Ed.) Erkenntnis Orientated. (pp. 287-303). Dordrecht: Kluwer.

Mayer, V. (1992). Carnap un Husserl. In D. Bell & W. Vossnkuhl, (Eds.). Wissenschaft und Subjectivität. Berlin: Akademie Verlag.

A Journey to Platonism with Edmund Husserl — 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.

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