# 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.

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

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

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

## What is Logic?

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

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

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

Husserl will beg to differ on both accounts, and in the "Prolegomena of Pure Logic" he tells us why logic is ** theoretical: **and by this term he does

**mean that logic is speculation, it means that logic does not tell us how we ough to think, but tell us formally what**

*not***.**

*is*## When Sciences Go Bezerk

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

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

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

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

But what happens when a particular field of science is ** too** broad? What happens if a field occupies issues of another field? What would happen if zoology would incorporate something like botany? Everyone would agree that there is what Husserl would call, in Greek, a "μετάβασις εἰς ἄλλο γένος" (Isn’t Greek pretty? It is pronounced "

*metábasis eis allo génos*") or a "trangression to another genus (field)". Botany is about plants, not animals … therefore it should never be considered a branch of zoology (the science of animals). In this case, botany is a field in its own right.

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

## Empirical Consequences of Psychologism

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

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

Here is another passage:

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

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

~ (A & ~A)

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

John Stuart Mill, it is well known, held the principle of [no] contradiction to be ‘one of our earliest and most familiar generalizations from experience’. Its original foundation is taken by Mill to be the fact ‘that belief and disbelief are two different mental states’ which exclude one another. This we know — we follow him

verbatim— by the simplest observation of our minds. And if we carry our observation outwards, we find that here too light and darkness, sound and silence, equality and inequality, precedence and subsequence, succession and simultaneity, any positive phenomenon, in short, and its negation, are distinct phenomena, in a relation of extreme contrariety, and that one of them is always absent when the other is present. ‘I consider the axiom in question’, he remarks, ‘to be a generalization from all these facts.’

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

Husserl is right: belief (defined as a mental state) is not a proposition, sound and silence are not propositions, light and dark are not propositions, and so on. But here is Husserl’s point: how can John Stuart Mill derive an ** absolute, necessary, universal** logical proposition, from

**experiences? What process leads us from one to the other? How can a logical law be a generalized statement from our physical experience in this world? Mill never says how this is so. This is precisely what David Hume criticized about induction.**

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

## First Consequence of the Empirical Supposition in Psychologism

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

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

## Second Consequence of the Empirical Supposition in Psychologism

Another problem that we have is that psychology is an ** empirical** science, therefore, all of its laws are known by contrasting them with experience. This is not the case in logic, whose rules are known

*a priori*(this means that these rules are known through reason alone, with no reference at all to experience).

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

All knowledge is only probable.

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

The statement ‘All knowledge is probable’ is probable.

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

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

.

.

.

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

And even if psychologists wanted to make logical laws as ** natural** laws, we have to ask, how is this statement justified at all at any level of psychologistic literature? The reason for this confusion is that many psychologists actually confuse the

*causal*laws of nature with the

*non-causal*logical laws, even though they try the best to derive one from the other.

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

## Third Consequence of the Empirical Supposition in Psychologism

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

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

If all As are Bs

If all Bs are Cs

———————–

Then: All As are Cs

So far so good! Well … may I ask you, my dear reader, where is the "ough" part of this logical rule? The word "ough" is nowhere to be found! For Husserl, this is a ** theoretical** rule … it tells us what

**, not how we ough to think. Now if I said something like: "If it is a fact**

*is***that all As are Bs, and that all Bs are Cs, then you**

*for you***to**

*ough***that all As are Cs." … then**

*think***statement is normative and**

*this***tell us how to think. This statement**

*does**does*have psychological content.

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

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

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

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

**.**

*the 3 itself* Combine all of these three consequences of psychologism, and what do you have in the end? Something very simple. If psychologism is true, then no knowledge is possible… and if ** this** is true, then we are led to skepticism. That will be the subject of our next blog post.

## References

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

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

## 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).

## Introduction

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

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

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

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

Dear Doctor,

…

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

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

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

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

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

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

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

## A Priest and the Birth of Semantics

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

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

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

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

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

Bolzano says something interesting:

[Objective representations] are not to be found in the realm of the real. An objective representation does not require a subject but subsists, not indeed as something

existing, but as a certainsomethingeven though no thinking being may have it; also, it is not multiplied when it is thought by one, two, three, or more beings. . . . Fort his reason, any word, unless it is ambiguous, designates only one objective representation (Coffa, 1991, p. 30).

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

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

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

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

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

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

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

**of the proposition. To give you an example:**

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

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

## Husserl and Bolzano

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

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

As we shall see here, as in the case of Bolzano’s "objective representations", ** senses or meanings are themselves totally abstract but objective**. They are not found in space or time, but in a sense

**independently of the mind or the physical world. They are not the physical sign, nor do they belong in the psychological activity of the brain. If I say: "Kennedy was killed in 1963", its meaning, its proposition,**

*subsist***true, and**

*is***, even if everyone believed it to be false. There is no better example to show that senses or meanings are**

*will always be true***. Psychologism’s error was to believe that they were not independent. As we shall see in a future blog post, this will lead to inevitable contradictions.**

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

### Sense (Meaning) and Referent of Proper Names

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

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

*Example 1*:

"The victor in Jena"

"The victor in Jena"

** Both proper names are exactly the same**, because we are using the same

**or physical represenation in graphic writing. Hence, they contain the same message (sense or meaning), and refer to one and the same object: Napoleon Bonaparte.**

*sign**Example 2*:

"The victor in Jena"

"El vencedor en Jena"

Here ** both proper names differ**, because the graphical and written representations or signs are different. One of the phrases is in English, the other one is in Spanish. Yet they both carry

**, because they tell us the same thing in both languages. They still refer to the same object: Napoleon Bonaparte.**

*the same sense or meaning**Example 3*:

"The victor in Jena"

"The defeated in Waterloo"

Here, both proper names differ, but there is now a ** difference in sense or meaning**. They both

**to Napoleon Bonaparte,**

*refer***. Why? Because both of these proper names are giving us two very different informations about the guy. Since their logical content is different,**

*but not in the same way***.**

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

*Example 4*:

"The morning star"

"The defeated in Waterloo"

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

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

## Sense (Meaning) and Referent of Universal Names

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

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

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

———————————-

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

## References

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

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

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

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

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

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

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

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

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

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

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

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

## Introduction

It is 1892, and Husserl was going to adopt a very unusual position in philosophy … one not very popular among philosophers. In fact, not even popular among mathematicians. He had to be honest, you know … but he had mixed feelings about it. Husserl’s change of mind in the start of the 1890s implied that he would have to be true to a dear teacher and friend, Franz Brentano, and tell him about how useless his view of geometry is.

Psychologism went through a Kantian path regarding geometry. For Immanuel Kant, there is an intuition of space which has four essential traits:

- Kant conceived space as a form of intuition, an
*a priori*condition for objects appearing to us as three-dimensional. - He conceived space the Newtonian way (as opposed to the Leibnizian). Leibniz’s conception of space is relational; for space to exist, there must be objects which can be related spacially or occupy space … no objects, no space. Newtonian conception of space on the other hand is absolute, existing with independence of objects themselves.
- He conceived space as Euclidean, and that is the
*only*way it can be given in experience. - Geometry as pure mathematics (
*mathemata*) can be constructed in pure intuition: as he called it "constructions from concepts".

Psychologism thought geometry as being the result of abstraction from experience, which is intuitively a three-dimensional euclidean space.

Yet, in his letter to Brentano in 1892, Husserl expressed him his change of mind. No longer was pure geometry to be considered a mere abstraction from experience itself, no longer about generalizations of objects of experience. Geometry is a science in its own right, ruled by all sorts of *formal* laws. And not only that … Husserl recognized that there was no *a priori* reason to restrict ourselves to euclidean geometry. Non-euclidean geometry is *just* as legitimate in all of its forms. Euclidean space is just *one* of an infinity of possible spaces.

This is indeed ** not** a popular position.

## What is Euclidean Geometry?

Simply speaking, euclidean geometry is the sort of geometry developed by Euclid, the famous ancient mathematician. It considers three-dimensional space as perfectly "flat" so-to-speak. If we translate three dimensions into two dimensions, we could represent space as a flat surface.

The geometry you learned in high school is precisely such geometry. In this sort of space, the angles of squares are all right angles, the Pythagorean Theorem applies to all right triangles. In this kind of space, if you add the three angles of a triangle they are equal to 180⁰ (not more and not less).

## What is Non-Euclidean Geometry?

It is not obvious to everyone that there can be other sorts of spaces than those we are experiencing intuitively right now, nor those we learned in high school.

The problem from the very beginning had to do with what many mathematicians historically called the ** axiom of the parallels**. It basically states that for any given line

*a*, and a point outside of that line, there can only be one, and

*only one*, line

*b*which is parallel to line

*a*. (Remember, lines extend to infinity, and

**lines will never intersect at any point of infinite geometrical space).**

*paralell*Interesting, isn’t it? For millenia, this has always been an issue in geometry. At face value, it seems that this is self-evident for many people in geometry. All self-evident truths in mathematics are called ** axioms**, which basically means that they are so extremely evident (almost to the point of exclaiming "duh!") that they do not need any proof at all. Yet, many other mathematicians did not regard this "axiom" of the parallels as self-evident, and stated that there was a need for mathematical proof.

Many people engaged in the search for such a proof throughout history. For instance, during the seventeenth and eighteenth century, a Jesuit priest called Gerolamo Saccheri (1667-1733) used a very important approach widely used in philosophy and mathematics: the use of *Reductio ad Absurdum*. How does this procedure work? Let’s say that you want to prove that the axiom of the parallels is necessarily and universally true. To do it, you begin by supposing *the opposite*: we are going to suppose (just for the sake of the argument) that the negation of the axiom of the parallels is true. If in the process of logical deduction from that, it produces a contradiction, then the negation of the axiom of the parallels is regarded as "absurd" (i.e. leads to contradictions), and, therefore, the axiom of the parallels is true.

Saccheri applied this procedure … he supposed the negation of the axiom of the parallels and derived from there using logical and mathematical rules. Alas! He did ** not** find a contradiction. Quite the opposite … he found that the negation of such "axiom" was

*perfectly consistent*. Contrary to what he expected, Saccheri accidentally proved that non-euclidean space is logically possible. He tried to save his views of the inherent and necessity of euclidean space saying that only such a space is intuitive: it is the only

**one.**

*real*

Carl Friedrich Gauss (1777-1855)

Yet, this final judgment by Saccheri did not stop a mathematician called Carl Friedrich Gauss from exploring the mathematical possibilities open by Saccheri’s accidental conclusions. If the axiom of the parallels is not an axiom strictly speaking, then that means that euclidean space *is not the only* mathematically valid sort of space there is. He wanted to explore other possibilities negating the so-called axiom of the parallels. He discovered that it was perfectly possible that in many ways *more lines* can pass through point *X* which are parallel to line *a*.

But this discovery was further elaborated by János Bolyai. Bolyai’s father, who happens to have been Gauss’ friend, tried to prove the self-contradiction of non-euclidean space, but to no avail. On the other hand, Bolyai (1802-1860) and, simultaneously and separately, Nikolai Lobachevsky (1793-1856) elaborated what is known today as ** hyperbolic geometry**, a set of non-euclidean spaces where space has a particular shape, my like a horse’s saddle.

Yet, there was this other mathematician, also obsessed with non-euclidean geometry called Gerard Bernhard Riemann (1826-1866) who developed another kind of non-euclidean geometry called ** elliptic geometry**. One of the possible elliptical spaces is a spherical-shaped space.

Both of these sorts of spaces are inherently different with respect to euclidean space, and different among each other. Let’s see some of these differences. For instance, in a hyperbolic space, if you have a line *a*, there can be multiple parallel lines which can pass through any point *X*, outside line *a*. In an elliptic space, there can’t be parallel lines, because two straight lines will always intersect, no matter what. In euclidean space, there can only be one parallel line.

Another remarkable difference between spaces is the addition of the angles of a triangle. In Euclidean space, if you add up the angles of a triangle, they will always give you 180⁰; but in hyperbolic space, the result will be less than 180⁰; and in the case of elliptic space, it will always be more than 180⁰.

There is another very interesting aspect to non-euclidean spaces, and it was pointed out by Riemann himself. Imagine, for instance that you are standing on the Earth … like you usually do. Haven’t you noticed that it seems that the Earth is flat? In fact, many people in ancient times did believe that. If you measure one small area like New York City, you are going to confirm that the Earth is flat, but as you increase the size of the measurement to, let’s say, from New York City to the Earth’s equator, you’ll notice that the Earth is no longer flat, but *curved*.

So, in many small areas of a spherical space, the mathematical truths of euclidean space and those of elliptic space are almost identical, the differences can be negligible for all practical purposes …. just as we know that the Earth is a sphere, but for all practical purposes, we can neglect this fact if we want to measure the area in our backyard. Riemann pointed out that there can be different degrees of measurements this way.

Also, we have to take into account that multi-dimensional euclidean and non-euclidean space can be measured and calculated mathematically. In Riemann’s time, it was thought that physical space was an euclidean three dimensional. Yet, mathematically speaking, four-dimensional space, or five dimensional, or even million-dimensional, or *n-*dimensional spaces are possible.

All of this means that physical euclidean space as we know it is just *one* out of many possible spaces. The measurement (metric) of physical space can actually be determined empirically (by experience).

This constitutes Riemann’s notion of ** manifolds**, we can take different "regions" of space and determine their metric properties. Husserl, inspired by Riemannian manifolds, would generalize this notion to the whole of mathematics as a

*mathesis universalis*. In its supreme form, Husserl would consider mathematics as a theory of manifolds, where certain "regions" of mathematics can be determined by positing axiomatic rules (or eliminating some of them), and positing sorts of numbers or mathematical concepts, and derive all sorts of consequences out of them as long as

*logical consistency*is preserved. That is why a mathematician is free to posit fractions, negative numbers, negative roots, decimals … you name it, and out of them derive a completely consistent mathematical theory.

## Consequences of the Adoption of Non-Euclidean Geometry

As I said before, Husserl’s position was a philosophical oddity at his time. As far as philosophers in general knew, mathematical obsession with non-euclidean geometry was a waste of time. The vast majority of philosophers (and mathematicians) believed that Saccheri was right: even if non-euclidean spaces were mathematically possible, there was absolutely no reason why we should explore them.

As far as these philosophers go, mathematics is a *technique*, not a theory itself about anything real. It is merely a *means* to an end, and it should develop all of those mathematical areas that *evidently* will be useful for us … we have to discard the rest to the waste basket as some sort of aberration. This was the same position held by many psychologists (i.e. proponents of psychologism) at the time regarding logic. Logic is the *art* of correct reasoning. Reasoning is a psychological process. Therefore, logic is a psychological *technique* which should be developed as long as it helps us think clearly, from a psychological standpoint. Forget about treating logic as a theory in itself … or else it will become as useless as non-euclidean geometry. Frege was furious at such attitudes!

After 1890, Husserl discarded all of these opinions to the waste basket. His mathematical side just wouldn’t allow it. Pure logic is not a technique, it is a formal theory which is a pre-condition for any other deductive theory used by science and other fields. Formal logic is a field of its own (not a branch of psychology): in other words, as he would say, "pure logic is a theory of all possible forms of theories" or a "theory of deductive systems". Mathematics was not a *technique* either, it is a theory of manifolds in its supreme form, all objectualities in the universe ** have** to be understood according to mathematics, not the other way around.

And essentially, that was what Husserl tried to say to his dear teacher and friend, Franz Brentano. Just like Bernard Riemman, János Bolyai, and Nikolai Lobachevsky, Husserl broke Kantian tradition of euclidean space as the only valid space in which we can obtain knowledge.

## Ironies of History

Henri Poincaré (Left); Albert Einstein (Right)

Another famous mathematician was Henri Poincaré (1854-1912), today considered by many as one of the fathers of the general theory of relativity. Why would that be? He made bunches of contributions at the time to special relativity, but the big one had to do with the whole idea of non-euclidean geometry. He was not a platonist, but a constructivist. For him, mathematics are about constructions of the human mind.

He realized that non-euclidean geometry was perfectly legitimate, and opened the door to the possibility of a non-euclidean theoretical explanation of the physical world. However, contrary to many philosophers and scientists of the time, he did actually open the door to the idea that experience may determine that a particular non-euclidean space may serve as basis for a *better* explanation of phenomena in the physical world than euclidean geometry. Unfortunately, his constructivist prejudices led him to believe that this would never happen because "obviously" every-day space is euclidean.

Albert Einstein (1879-1955) *did* read Henri Poincaré, and had an idea to solve a scientific problem. If space-time itself (space and time as one entity) is physical, not just a pure nothing, and the velocity of any inertial reference frame affects the measurements of mass, distance, and time, then that might suggest a different conception of space and time as they have been traditionally treated in history.

According to special relativity, nothing can travel faster than light’s velocity, something which left a very big hole in Newton’s conception of gravity. According to Newton, the gravitational influence of one mass to another is instantaneous. So, if the sun disappeared all of the sudden, the Earth and all of the other planets would fly away instantly. This is impossible according to special relativity, because the velocity of gravitational effect *should* have the velocity of light as its limit.

Under these circumstances, Einstein had two choices:

- Either assume that space-time is a very simple euclidean space, while it unnecessarily complicates the scientific theory to explain gravity.
- Or assume that space-time is non-euclidean, which would complicate the geometrical model, but it would
*simplify*considerably the explanation of gravity in light of special relativity.

He chose to assume that space-time is non-euclidean, which helped him suppose that space-time is itself *curved* by the presence of mass and energy. The reason why you are stuck to the ground (gravitationally speaking) is not because the Earth is "pulling" you towards it as Newton supposed, but rather because the Earth’s mass creates a four-dimensional space-time distortion around it like the following illustration, and you are *literally* sliding downwards in that space-time. (In the following illustration, the three-dimensional space-time curves are represented as a two dimensional surface around the Earth).

(Illustration courtesy of Johnstone)

The Earth orbits around the Sun because, like a marble over a curved surface, the Earth moves along the curve of space-time created by the massive presence of the sun. If the sun disappears, though, it would create a space-time wave whose gravitational influence would not reach us until about 8 minutes later (the time it takes light to reach us). Einstein, then, was able to provide a better explanation than Isaac Newton for phenomena which Newton accounted for, plus more: he could also explain why light bends in a certain way when it is close to a massive object, the Doppler Effect, and the second twin paradox.

In its attempt to stick to the sensory experience, psychologism was a disaster. Its insistence to stick to euclidean space because that is our empirical experience got the best of it, especially those philosophers who were led by Kantian and Neo-Kantian philosophies. Frege complained that a psychologistic view of logic had actually *stalled* logic’s development, and he was quite right. If you say that logic is a *technique* to think well in a psychological level, then any logical discovery which deviates from that does not serve to that end, and would be regarded as useless. Husserl also believed that, although mathematics was far more developed than logic (he called it "logic’s fat sister"), the irruption of psychologism in mathematics would be a transgression to a completely different field. If mathematics is a *technique* to understand the world, then we should prevent further development in areas that some people at one time would regard as "evidently useless".

Even during the 1980s, antiplatonists such as Phillip Kitcher would be so annoyed by what he considered "useless fat" of mathematics, that he said that we should cut it and throw it to the waste basket.

Imagine what would have happened to science, especially to Einstein and special-relativity, if mathematicians would have actually carried out what these science-loving people suggested.

: This article serves to refute one of the most widespread prejudices in analytic philosophy since W. V. Quine’s essay "Two Dogmas of Empiricism". For Quine, there is no distinction between formal and natural sciences (analytic or synthetic judgments), because they constitute a unitary whole which can be revised in light of recalcitrant experience; hence, recalcitrant experience can revise logic and mathematics. This conviction was further reinforced by Hilary Putnam’s statements that there have been instances where logic and mathematics were revised because of experience, he mentions the general theory of relativity’s revision of geometry as one of those instances.Final NoteIf you have been paying attention, you have to realize that mathematical revision took place, but not because of "recalcitrant experience", but because of a problem

the realm of mathematics: the problem of the so-called "axiom of the parallels". Within mathematics, non-euclidean geometry was always mathematically valid even before the success of the general theory of relativity. Einstein never revised non-euclidean geometry, he only adopted it as a mathematical model on which to build his scientific theory, and he succeeded. Similar events have happened in science, for example, Hilbertian spaces have served as basis for many areas in quantum physics, or the way chaos theory and fractals have served as mathematical models to explain many phenomena in the world.within

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