# 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

everythingis 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 beactualis 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

*, 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:*

**matters-of-fact**: Which include*Analytic-a priori statements**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.: These are always true, but they cannot be formalized**Synthetic-a priori statements***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 “

*“. 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.*

**formal-ontological categories**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

*entities. This is due to the fact that mathematical statements being*

**self-subsisting***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.

# References

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.

=-=-=-=-=

*Powered by Blogilo*

In 2006 I published this book for the first time, and I’m proud to say that this is the fourth edition of *The Relation between Formal Science and Natural Science*. In this book, I use Edmund Husserl’s philosophy of logic and mathematics, as well as his semantic doctrine, in order to understand the nature of formal sciences. It posits the existence of ideal meanings and mathematical objects, which are themselves a condition of possibility for any truth and any science whatsoever. It advocates for the search for a criterion to determine the distinction between analytic and synthetic judgments, while rejecting Quine’s arguments against it. At the same time it rejects several antiplatonist options such as Mario Bunge’s fictionalism, and Karl Popper’s semiplatonism, while proposing Husserlian epistemology of mathematics as an alternative, which is essentially a sort of "rationalist epistemology" as Jerrold Katz suggested. Finally, the book criticizes the Quine-Putnam theses, especially the one which states that logic and mathematics can be revised in light of recalcitrant experience. Usually three cases for such revision are constantly presented in this debate: quantum logic, non-euclidean geometry and the general theory of relativity, and chaos theory. I show that none of these *a posteriori* matters-of-fact have revised any *a priori* formal fields such as mathematics and logic.

The book’s website has also undergone major surgery, changing it from plain HTML to a Drupal platform. This is how it used to look like:

(Click for Larger Version)

This is how it looks like:

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.

=-=-=-=-=*Powered by **Blogilo*

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:

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

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

**to be.**

*ought* 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

**courageous serves as the foundation of the norm that all soldiers**

*is***courageous.**

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

(*a* + *b*) (*a* – *b*) = *a*² – *b*²

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;

. 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. …it falls to logic to discern the laws of truth. 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.From the laws of truth there follow prescriptions about asserting, thinking, judging, inferring. … In order to avoid any misunderstanding and prevent the blurring of the boundary between psychology and logic,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(Beaney, 1997, 325-326).I assign to logic the task of discovering the laws of truth, not the laws of taking things to be true or of thinking

What is Husserl and Frege’s point here? Very simple, the norms which we should follow ** if** we wish to find the truth are

**founded on psychological laws, but rather in**

*not***. These theoretical laws themselves tell us nothing about mental processes or physical or biological laws which operate the brain or the mind.**

*theoretical logical laws***.**

*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

processes. Therefore, logic belongs in psychology.psychological

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:

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

The proposition expressed in "JFK was killed in 1963" ** is true** and

**, 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**

*will always be true***of our minds.**

*independent* 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

**regardless of whether tomorrow we were to believe psychologically that "2 x 2 = 5".**

*proposition "2 x 2 = 4" is true and will be forever true*From these facts we have to make several distinctions:

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

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

### Third Prejudice

Psychologism’s third prejudice goes as follows:

If we find a logical or matheatical proposition to be true it is because we find it

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.evident

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

**side of logic. We find the principle of no-contradiction as evidently true for one reason only: because it**

*theoretical***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**

*is**evidently*true that there

**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**

*is***truth (which escapes all psychological representations) that there is a trillionth interger of the number pi. …. We don’t know what it is, but**

*ideal***it**

*ideally***there, it exists. We know this**

*is**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:

: 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*Ideal*, these truths do not change or are affected by temporal events, in a very real sense they are eternal.*atemporal*: By "real" Husserl does not mean "existent". Ideal entities, meanings and relations do exist independently of us. What Husserl means by real [or "*Real**reell*"] is that it is concerned with particular moments in time. This is the realm of physical objects, which persist and change in time, and also of psychological acts which also occur in time.

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

## References

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

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

## Hume’s Big Skepticism

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

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

From the point of view of knowledge, the question is, what are we *really* given in experience? For Hume, all that we are given from the "outside" world are what he calls ** impressions**: what we see, smell, taste, hear, and touch. For Hume, we are not given actual objects, just impressions. One

*possible*explanation for our impressions is that there are actual objects out very much like the way we sense them. This itself is a matter-of-fact. I remind you that, as a matter-of-fact, this is a just

**, and**

*one possibility***. There can be vast infinite possibilities of explanations for the impressions we have.**

*not the only one* 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)

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*is*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*true*true, and another thing is*being*true. Regardless of any species’ constitution, if a species believes that something is true or false, does not mean that it*holding or believing something as*true or false.*is*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:

to any species of rational beings able to grasp it, even if it is for humans, for angels, or for the gods.*truth is one and identical* - 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

different things than what we mean, then it is a problem of the*mean*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*meaning*(i.e. propositions), and their fulfillment in a state-of-affairs (or "facts").*mean*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

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*be*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.*not* - 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"2×2=4" does*truth**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 trueit is fulfilled by a matter-of-fact, not that a matter-of-fact "derives" or "infers" a universal truth.*if* - 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,

is nothing more than a network of true propositions which are*truth**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
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?*truths*This would be obvious to everyone, but if we reflect a little bit about it, we become aware that our own ego and its psychological acts belong to this universe, which would also mean that every time I say "I exist" or "I have such and such experience", it would be instantly false in a truth-relativistic point of view.

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

## Relativism in General

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

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

## References

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

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

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

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

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

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

Pierre Duhem (1861-1916)

Duhem was *far* more careful than that, as philosopher of science Donald Gillies has pointed out. Pierre Duhem did recognize that ** in physics** … and only in

**… 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. …**

*physics* 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 . . . L*n*

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 . . . L*n*

C1 & C2 & C3 & C4 . . . C*n*—————————————-

P

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

## The Formal Components of Scientific Theories

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

### Acts of Meaning

The only question we never really answered in these series is how do we formulate propositions? Which acts of consciousness intervene in these process? Remember, our primal constitution is of states-of-affairs. For example, we can constitute a white sheet of paper on the desk. However, by other acts of consciousness, I mentally constitute another different sort of form *to refer* to that state-of-affairs. That mental act is what Husserl calls a ** meaning act**, which makes possible for me to say "There

**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**

*is**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

**taller than Mary".**

*is* And the word "is" in this context *does* correlate with a categorial form, but ** not** it is not a formal-objectual category … but a

**. 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**

*meaning category***. Let me give you an example of what I mean.**

*meaningful manner* 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***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**

*grammar rules**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

**. We can see here the relationship between the two … but how do they integrate in a "**

*formal-objectual categories are the basis for mathematics**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***, where meaning categories arrange objectualities into meaningful propositions. This is called by Husserl a**

*grammar***(in other words, how meaning categories "shape" propositions). This stratum is ruled by**

*morphology of meanings**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

**. Here we find formal-objectual categories such as: cardinal numbers, ordinal numbers, sets, relations, parts-whole, and so on.**

*morphology of formal-objectual categories*### 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

**. 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.**

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

On the side of mathematics, mathematics becomes a ** theory of manifolds**, a

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

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

## Some Interesting Facts …

Rudolf Carnap is known to have made the distinction between ** formation rules** and

**, and this went through history of logic as being**

*transformation rules**his*particular contribution to the subject. But we know as a matter of fact that Carnap was pretty much familiar with Husserl’s

*Logical Investigations*, and used Husserlian terminology extensively in both of his first major philosophical works:

*On Space*and

*The Logical Structure of the World*. However, due to his relationship with members of the Vienna Circle who were pretty much anti-Husserl, he wanted to water down Husserl’s contributions to his philosophy, especially in

*The Logical Structure of the World*.

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

Wadda ya know!

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

## References

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

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

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

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

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

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

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

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

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

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

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

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

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

## Problems with Sets

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

Set theory had two major paradoxes at the time.

### 1. Cantor’s Paradox

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

*S*:

*S* = {*x*, *y*, *z*}

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

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

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

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

Naïve set theory states that it is possible to form the

. If it is the "set ofset of all setsallsets", 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 definitionthan 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.greater

### 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

**, no matter what. Logical and mathematical schemes are a network of**

*necessary and always true or always false***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**

*essential***(not only formal) relationships among these concepts, and science also depends on this.**

*material*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

**to account for our knowledge of essences and formal categories.**

*has*### 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:

**(sensible content), and**

*material content***(formal-objectual categories). How are these constituted?**

*formal arrangement* 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:

: 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 Intuition*, 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 perception*(my constitution of objects in my imagination: fairies, imaginary tables, crystal castles, and so on).*sensible imagination*: 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).*Categorial Intuition*

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:

**. 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**

*mixed-categorial acts***, or**

*three pencils***, or**

*a set of books***, and so on. All of these constitutions of sensible plus formal states-of-affairs are the result of mixed-categorial acts.**

*Megan being taller than Mary*### 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

**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 "**

*the**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

**, with which we constitute pure categorial forms, they become our object of our consciousness. This is also called**

*pure-categorial act***.**

*formalization*So, when we talk of ** mathematical intuition** in Husserl’s case, we are talking about

**plus**

*categorial intuition***. That’s how we know abstract objects such as numbers, sets, and so on.**

*categorial abstraction*### Eidetic or Essential Intuition

Yet there is a third form of intuition which Husserl calls ** eidetic** or

**: an intuition of essences. What does this intuition do? It gives us essential relations at once, we are able to recognize through understanding the**

*essential intuition***and**

*universality***of what is being given. Let me give you a very obvious instance of how this happens:**

*necessity*"*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

**recognize that it is universally and necessarily valid. We don’t have to go through a tedious process of substituting**

*immediately**x*and

*y*with different numbers to check if there is some arithmetical exception to this rule. No! Not at all! We

**that there are**

*know***exceptions to this mathematical truth.**

*no*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. InExperience and Judgmenthe 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
logic and mathematics are necessary for science. He just argues, from the pragmatic point of view, that they work.*why* - He is unable to explain the success of the predictions made by logic and mathematics, especially in developments which initially seem pointless: as it happened with non-euclidean geometry, negative roots, Hilbertian spaces, among other mathematical developments.

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

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

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

## References

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

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

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

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

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

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

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

## Sense and Referent of Propositions (Judgments)

One of the most widely discussed issues in semantics has to do with the sense (meaning) and referent of judgments.

The best known initial proposal was Frege’s. For Frege, two assertive sentences can express two different propositions (he called them "thoughts"), which could refer to one sole object. For instance, if I have two assertive sentences like this:

- The morning star is a planet.
- The evening star is a planet.

then both of these sentences express different senses (different propositions), since "the morning star" and "the evening star" are two proper names which express two different senses, but refer to one sole object. And what is the referent of an assertive sentence? For Frege is its truth-value. Sentences (1) and (2) share the same truth value, therefore they are two senses which designate the same referent: truth.

Yet, no sooner has he proposed this semantic doctrine that we can come up with cases which show a very, very big hole in Fregean semantics. Let’s say that I can come up with two sentences like this:

- The morning star is a planet.
- Paris is France’s capital.

Intuitively we would feel very uncomfortable with the idea that both of these sentences have the same referent, since they talk about two different "facts" (to say it loosely), yet they refer to the same object: truth. ~ Scratching my head ~

For this reason, Frege’s semantics was not very popular, but it fit his own agenda of trying to prove that arithmetic can be derived from logic. Unfortunately he failed. Still, he held more or less this view until 1919, with his famous essay "Thought: a Logical Inquiry". By the way, I love this particular essay, it is of his best, and in terms of train of thought and arguments, it is beautiful … but that’s just me.

As I have said in an earlier post, Husserl did agree with Frege about the sense and referent of proper names. He started to disagree with Frege regarding universal names, since Husserl believed that their sense are concepts and their referents are the objects which fall under these concepts.

Yet, their differences couldn’t be more when it came to assertive sentences.

## States-of-Affairs

Husserl was not just a logician, mathematician or semanticist, he was also worried about how to develop an adequate theory of knowledge, searching for principles to explain how did we know stuff. Remember, he was also Franz Brentano’s disciple!

Yet, many people seem to forget that it was in the *Philosophy of Arithmetic* where he said that in some way we have an experience of *sets* or *groups*. Husserl’s change of mind indicated that he no longer wanted to reduce experience to pure sets, but he wanted to recognize all sorts of abstract formal categories as part of those experiences. These categories include:

- Sets
- Relations
- Part-Whole
- Cardinal Numbers
- Ordinal Numbers

So, for him, it were not merely sets, but also there were other ways sensible objects are formally organized.

See … in every experience we have about the world, there are two very different elements we must distinguish:

**Sensible Objects**: pencils, houses, notebooks, computers, and so on.**Formal Components**: ways of relating these sensible objects objectively.

Let me give you a particular example of how this is so. Tell me: What do we have here?

You might say: "Pencils". Nah! Look again. You just don’t have "pencils" here. You may constitute them in any the following ways:

- A
of Pencils*Set* Pencils*Three*,*First*,*Second*Pencils*Third*- Pencils
*in a Row* Red Pencil, and other*One*Pencils*Two*Red Pencil,*One*Green,*One*Blue*One*- …. 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

**the table, that your keyboard is**

*on***the desk, that the glass of water is**

*on***the keyboard (it which case … be careful not to drop it on the keyboard :-S), that the pizza is**

*beside***the mattress (well … this just applies to Oscar Madison in**

*under**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

**. They are objectual, because they are constituted**

*formal-ontological categories**at once*along with sensible objects. In fact, these formal-objectual categories are

**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**

*founded**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

**of pencils" and "**

*set***pencils" is precisely that one of them is constituted as a**

*three**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

**than Megan, I am constituting**

*shorter**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

**(and objectually) founded on sensible objects: Megan and Mary. So … according to Husserl … Megan and Mary comprise the**

*objectively***, which is the reference basis for two**

*situation-of-affairs***: Megan being**

*states-of-affairs**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

**whatsoever, which means that anything that exists must be arranged formally all of these ways.**

*being*## 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

pairof 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:

. 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).Many different statements or assertions may equally truly describe one and the same fact

By making a semantic distinction between states-of-affairs and situations-of-affairs, Husserl seems to have covered all the bases. For him, states-of-affairs are the facts referred to by propositions. At the same time, these states-of-affairs have situations-of-affairs as reference basis.

So, if we were to summarize Husserl’s doctrine of sense (meaning) and referent (objectuality), we would do it this way.

Finally, notice that Husserl doesn’t bow down to phenomenalism (the doctrine that we are actually given are sense-data: gradations of colors, sounds, tastes, etc.) For him, we are given objects *in a specific formal arrangement* (states-of-affairs), and all knowledge stems from them. For him, sense-data (he calls them *hyletic *data) are the result of processes of sensible abstraction. They are never primordially or evidently given first hand.

## References

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

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

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

Ifrah, G. (2000). *The universal history of numbers: from prehistory to the invention of the computer*. John Wiley & Sons.

Pinker, S. (1994). *The language instinct: how the mind creates language*. NY: Harper Perennial.

Popper, K. (1994). *Knowledge and the body-mind problem: in defence of interaction*. London & NY: Routledge.

Wynn, K. (1992). Addition and subtraction in human infants. *Nature, 358*, 749-750.

## A Letter to a Friend …

Carl Stumpf (1848-1936)

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

TO CARL STUMPF …

with Honour and Friendship

Very simple words, yet they express everything.

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

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

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

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

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

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

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

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

## "Imaginary" Numbers

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

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

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

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

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

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

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

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

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

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

-4 = (4) (-1)

Then you solve the square root of four, and represent the root of negative one as the letter

i. So, the negative square root of four, is identical with 2i. This would be an imaginary number.

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

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

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

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

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

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

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

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

You could say that negative numbers, fractions, etc. are a very clever use of numbers at the *symbolic *level, that they don’t refer to actual formal structures in objectualities at all, that they are just conceptual tools. Yet, Husserl was not a formalist. Numbers are not mere symbols or signs. Signs are the means to express them symbolically, but these *signs have meaning*. And ** if** these numbers work, then that means that they must

**to some formal structures which are constantly applied to real objects in our mathematical calculi. THAT would explain why do they work (and work so well!)**

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

If this is true … then two conclusions are inevitable.

## A Non-Reductionistic Approach

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

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

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

So, we are here at a very novel notion of mathematical objects. Many mathematicians and philosophers have tried (and many are still trying today) to reduce one kind of mathematical object to the other. ** Husserl rejected all of that**. For him, these

**structures (or**

*formal-objectual***, as he called them later), are all non-reducible to one another. These categorial forms include (but not limited to):**

*categorial forms*- Sets
- Ordinal Numbers
- Cardinal Numbers
- Irrational Numbers
- Negative Roots
- Parts and Whole

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

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

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

Imagine that!

## References

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

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

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

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

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

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

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

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

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

## Being Franz Brentano’s Disciple

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

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

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

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

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

*Psychology from an Empirical Standpoint*.

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

*x*is a fact, and it is, or if you say that

*x*is not a fact, and it is not, then you are telling the

**. Simple … right?!**

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

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

*Privatdozent*.

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

## Intentionality

As I said, Brentano was looking for an adequate correspondentist view of truth, and he thought he could do that from a psychological standpoint (psychologism). For that, he wanted to use the concept of ** intentionality**. What is intentionality? Very simple: it is that capacity of the mind to be

*directed*to a particular

**. An objectuality is the referent of our intentional acts, something our mind refers to. Intentional acts include but are not limited to: desiring, loving, hating, thinking about, be concerned about, and so on. Each one of these mental phenomena are**

*objectuality**intentional*: you desire

*something*, you love

*something*, you hate

*something*, etc. That "something" being referred to by these mental acts is a particular objectuality, the referent of my desire, love, hate. An objectuality can be real or imagined. I can

*desire*a crystal castle floating on the clouds, for instance … my desire is directed towards it.

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

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

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

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

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

*any*group of objects as sets: a

*set*of mountains, or a

*set*of students, or a

*set*of pencils.

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

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

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

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

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

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

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

If … I attempt to count the trees in a certain area of the park, I … must do something more than just be conscious of them, or even clearly see them… I must rather,

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

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

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

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

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

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

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

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

## So … Was Frege Correct?

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

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

If a totality of objects, A, B, C, D, is our presentation [

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

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

Certainly one distinguishes in complete generality the relating mental activity from the relation itself (the comparing from the similarity, etc.). but where one speaks of such a type of relating activity, one thereby undderstands either the grasping of the relational content or the the interest that picks out the terms of the relational content or the interest that picks out the terms of the relation and embraces them, which which is indispensable precondition for the relations combining those contents becoming observable. But whatever is the case,

one will never be able to maintain tha the respective act creatively produces its content. (Husserl, 1891/2003, p. 44).

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

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

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

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

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

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

## References

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

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

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

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

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

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

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

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

# A Journey to Platonism with Edmund Husserl — 1

(1859-1938)

## A Terrifying Frustration and a Definitive Turn

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

… In the case of the word "number", for example, the aim is to exhibit the appropriate presentation and to describe its genesis and composition. Objects are presentations [

Vorstellungen]. … Since everything is now a presentation, we can easily change the objects by now paying attention, now not. We pay less attention to a property and it disappears. … For example, let us suppose that in front of us there are sitting side by side a black and a white cat. We disregard their colour: they become colourless but are still sitting side by side. We disregard their posture: they are no longer sitting, without, however, having assumed a different posture; but each one is still at its place. We disregard their location: they are without location, but still remain quite distinct. Thus from each one we have perhaps derived a general concept of a cat. Continued application of this process turns each object [into a more and more bloodless phantom]. …… the difference between presentation and concept, between presenting and thinking, is blurred. Everything is shunted off into the subjective. But it is precisely because the boundary between the subjective and the objective is blurred, that conversely the subjective also acquires the appearance of objective. …

… In combining under the word "presentation" [

Vorstellung] both what is subjective and what is objective, one blurs the boundary between the two in such a way that now a presentation in the proper sense of the word is treated like something objective, and now something objective is treated like a presentation. Thus in the case of our author, totality (set, multiplicity) appears now as a presentation, now as something objective. …… According to the author [Husserl] a number consists of units. He understands by "unit" a "member of a concrete multiplicity insofar as number-abstraction is applied to the latter" or "a counted object as such". … In the beginning, the objects are evidently distinct; then, by means of abstraction, they become absolutely the same with respect to one another, but for all that, this absolute sameness is supposed to obtain only insofar as they are contents. … (Frege, 1894/1972, pp. 324-325, 331)

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

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

*also*believe this. And who is Gottlob Frege?

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

In it, he engaged against forms of ** psychologism**, and

**. In this context, "Psychologism" is the philosophical view that the validity of mathematical truths must rely in psychological processes of sensory abstraction, and "naturalism" means that mathematical concepts such as numbers, ultimately refer to natural or physical objects. There are other trends, such as**

*naturalism***, in which mathematical concepts tell us about aspects of experience;**

*empiricism***, in which numbers are constructed by our minds;**

*constructivism***, in which mathematical objects are socially constructed within a culture; and**

*anthropologism***where mathematical symbols are treated as signs and nothing more than that.**

*formalism* Frege proposed a ** realist** view of mathematics:

**means that mathematical concepts refer to existing abstract mathematical objects. The sort of realism proposed by him is called**

*realism***: which means that these abstract mathematical objects exist**

*platonism**independently*from the physical and psychological worlds.

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

## Husserl’s Background

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

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

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

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

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

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

## Reference

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

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

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

=-=-=-=-=*Powered by **Blogilo*