Thursday, February 24, 2011


image via here

There is a common difficulty in all interpretations of Kant's philosophy of mathematics, including my texts. To answer the question: What is an intuition? Even the researches that are related completely with the term 'intuition' start without a thorough definition of the term. So now, for a while, I have been trying to formulate in one sentence what an intuition is. First of all intuition Kant uses is a technical term and does not have the same meaning with the one that is used in English, namely  "(knowledge from) an ability to understand or know something immediately without needing to think about it, learn it or discover it by using reason" (Cambridge online dictionary). Kant uses Anschauung in German which means view or opinion, but the term is translated to English as intuition since Kant interchangeably uses Latin word 'intuitus'  (cf. Potter, Reason's Nearest Kin, p. 21). I know I still did not give a definition of it. And it will still take some more explanations to give a proper description or definition.

First, let us look how Kant uses the term 'perception'. When representation is accompanied by consciousness it is perception (Critique, B376, Bennett Translation) and when perception is a state of person it is sensation and when it is as perception of something it is cognition (ibid). Finally when a cognition is related directly/ immediately to an individual object, it is  intuition and when it is related indirectly and with general properties of several objects it is a concept (ibid). This is the warm up for the epistemic stuff although we will not use the terms 'perception'  and 'cognition' again. I just thought this paragraph gives a nice categorization in level base, and helps us to understand that concept and intuition are both representations of objects and are at the same level.

We are almost there... But let's understand a bit throughly what sensibility (Sinnlichkeit), the mere forms of sensibility  and sensation (Empfindung) are. Sensibility is "the name of the capacity for acquiring representations that reflect how we are affected by objects. So objects are given to us by means of sensibility, and that's our only way of getting intuitions" (Critique, B33, Bennett Translation) The mere forms of sensibility are present a priori in the human mind and these are a priori intuitions. These appear in the mind "even when there is no actual object of the senses and sensation" (ibid, B35). And "when an object affect us, it's effect on our capacity for representation is sensation" (ibid, B34).  Bennett notes in his remark to B35 that "sensation refers to the detailed content of what the senses dish up and the senses refers to every aspect of our capacity for passively receiving data". I haven't decided whether the difference between sensation and sensibility is important, but it is crucial to understand the difference between sensibility and the mere forms of sensibility since this is how Kant distinguishes empirical and a priori intuitions.

Now one more step: "If a representation contains sensation (which presupposes the actual presence of the object), it counts as empirical; if no sensation is mixed into it, the representation is pure. [Recall that ‘representation’ = ‘intuition or concept’.] We can call sensation the ‘matter’ of sensible knowledge; and what is left when that is removed is the ‘form’. Thus pure intuition contains merely the form under which something is intuited..." (Critique, B75, Bennett Translation)

Examples? Consider the representation "body", remove from it everything that belongs to the understanding: substance, force divisibility... (we eliminated the concept part), remove from it everything that belongs to sensations: impenetrability, hardness, color... what is remaining: extension and shape (cf. ibid B 35) This is pure intuition. But, note that "[it] is just a fact about our nature that our intuition can never be other than sensible, all there is to it is our being affected by objects in a certain way." (ibid B75)

So what is an intuition? An intuition is a singular and immediate representation of an object, when the object is given to us by sensibility. Why Kant does not call it merely an object than? Because objects are out there, they act on the senses and the reason thinks about them, but what's an object in the human mind? It can only be represented and it is either a concept or an intuition, depending on the faculty (sensibility/ reason -more correct term would be Understanding-) that deals with it.

Then what is a pure or a priori intuition? A priori intuition is the mere forms of sensibility. You cannot say the form of intuition is a priori intuition since intuition comes represented through sensibility whereas a priori intuition resides a priori in human mind. And what can be mere forms of sensibility? These are primarily space and time, and then comes the a priori intuitions such as extension, shapes... Kant at some point believed that numbers were a priori intuitions too (cf. Kant, Inaugeral Dissertation,  §23, referred by Potter, Reason's Nearest Kin, p. 42) but let's not get into the number talk here since it's a whole another story, which hopefully I will deal with later.

Review of Hintikka's text? Coming up after a bit more clarifications...

Monday, February 21, 2011

Before digging in more: what's with this analytic/synthetic distinction?

When I was reading Hintikka's "Kant on the Mathematical Method" in Posy's collection , I had a difficulty to structure a good distinction between analytic and synthetic method, where Hintikka mentions briefly in pp. 30-31. Accordingly analytic method goes back to Plato and gets stronger with Descartes where there are no constructions used and it proceeds with assuming the result and going backwards. Synthetic method tries to effect the desired result and it's based on the use of actual constructions. (cf. Posy, p. 31) As far as I understand, the latter became as a proof method in Euclidean/ synthetic geometry as well while the analytic method remained as an old tool after the development in analysis in 19th century (Here it says reductio ad absurdum is that kind of a method but I doubt it). Descartes used the analytic method in Discourse on the Method and in Meditations since he wanted to reach certain new truths. In analytic method the proposition that is used in the beginning is not like a theorem but more like an hypothesis. The premises are to be found during the process, they are not assumed in the beginning. Once premises, axioms and inference rules are found for the desired hypothesis then synthetic method can be used to prove that the general proposition hold. 

Then there is also the talk of analytic proof. Is this the same thing with the analytic method. NO. By definition, an "analytical proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not make use of results from geometry". Synthetic proof is the same thing with the synthetic method.

Then what is analytic philosophy? Does this have anything to do with the methods or proofs used? Yes but it is not related to any of the above methods or proofs. Analytic philosophy was developed after formal logic got really strong and the clarity of arguments can be checked by formulating the methods and the arguments logically. 

Then what is analytic/ synthetic distinction for judgments and proposition.  This is whole another story. There are at least three main different definitions of analytic and synthetic belonging to Kant, Frege and Quine. 

For Kant in analytic judgments the concept of predicate is contained in the subject. These are affirmative ones. In synthetic judgments the predicate is completely different from what we think in the concept of the subject. (A7/B11) His examples are:

  • "All bodies are extended"  (analytic)
  • "All bodies are heavy" (synthetic)
In the first case all I need to do is to analyze the concept "body" to reach to the concept "extended". Hence the concept extended is contained in the concept body but this is not the case in the second example. These examples use mere syllogistic logic, no strict object and concept distinction, no quantifiers...

How does Frege define analytic./ synthetic distinction? According to him when a proof of a true proposition is carried on by purely logical means and when the premises and definitions of the terms can be given logically then this proposition is analytic. If one has to use in the argumentations of a true proposition, rules belonging to some special science apart from logic and/or terms that are not logically described then this proposition is synthetic.(Grundlagen, paragraph 3) Frege's definition uses two different notions to define the same concepts, a new kind of of logic: a very strong predicate logic, and propositions instead of judgments. While Frege is concerned with the justification of the propositions Kant is concerned with judging synthetically and with judgments. This last sentence will be of importance also when we discuss discrepancies in Hintikka's paper.  

And lastly the distinction of Quine, the contemporary consensus of the analytic/ synthetic: In "Two Dogmas of Empiricismhe defines analytic propositions as based on the meanings and independent from the facts. And the synthetic propositions as the propositions that are grounded on the facts. This paper of Quine is beautiful, the structure, the method, the feeling of it is really something. After arguing impressively whether there can be any distinction between analytic and synthetic propositions he concludes that " For all a priori reasonableness, a boundary between analytic and synthetic statements simply has not been drawn. That there is such a distinction to be drawn at all is an unempirical dogma of empiricists, a methapysical argument of faith" ("Two Dogmas", p. 37)

Any other uses of analytic and synthetic? Let me think a bit more... 

Friday, February 11, 2011

Kant's Philosophy of Mathematics

On the way of completing my first chapter of the Ph.D thesis by mid April, I started to read Posy's collection of essays on Kant's Philosophy of Mathematics. It's the best compilation in my area including papers from Hintikka, Parsons, M. Friedman, Melnick, Posy and by many others. I do not know if there is a *right* interpretation of Kant in terms of his philosophy of mathematics. His scholars could not even agree what Kant means by "Intuition" in analyzing mathematical objects and methods. There is a big debate started in the mid 60s between Parsons and Hintikka and more work followed by other scholars based on this debate. Hintikka, in his paper "Kant on the Mathematical Method" refers Kantian intuitions as particulars and claims that there is nothing intuitive about intuitions (p.23, Posy). He uses Kant's definition in Logic: "Every particular idea as distinguished from general concepts is an intuition". (ibid) He refers to other sources which carry the similar meaning: Kant's Dissertation 1770 Section 2 Paragraph 10, Critique A320/ B376-7 and Prolegomena  Paragraph 8

He claims that if we read Kant's Transcendental Aestetic in the Critique, the intuitions are referred as "mental images or an image before our minds eye" (p.26, Posy) but then he argues that in connection with Kant's mathematical views the definition of intuition which should be taken into account is the one at the end of the Critique, in the Transcendental Doctrine of Method. Then we will have no problems with Kant basing Arithmetic and Algebra on intuition, he claims. Otherwise, if the intuitions are taken as mental images, there is no way to interpret Arithmetic and Algebra being based on intuitions (ibid). Hence according to Hintikka, singularity of intuitions are essential and immediacy is a by product of singularity. It follows that intuitions, then can be given logical formulations with instantiations and existential quantifier eliminations . 

Parsons disagrees and argues that immediacy of intuitions are essential and in interpreting Kantian intuitions for his philosophy of mathematics, one needs more than logic, a method similar to perception. 

I am halfway with the Hinttikka's essay. So, let's dig in and see how  he expresses these logical formulations of intuitions. 

image from here

Friday, December 18, 2009

Transcendental Deduction A

These are my resuls of studying the deduction A.

 A summary...
Transcendental process:
In order for the transcendental unity of apperception (one’s a priori knowledge of the identity of self), pure imagination is needed (production, uses rules from categories). For pure imagination to work sensibility is needed (order is given by time).
Empirical process:
In order for the object of sensibility (passive intuition), imagination is needed (reproduction). For imagination, unity of apperception is needed (one’s knowledge of experience and recognition).
Objective validity of categories comes from the transcendental function of faculties. Pure understanding which also shelters categories, ascribes the rules for the pure imagination and also needs pure imagination for the activation of the concepts. Pure imagination needs the form of sensibility (time), which is objective and a priori. Given that the rules of understanding are priori and the appearances that are reproduced that activate the concepts are based on an objective form (time) a priori concepts of understanding (categories) are objectively valid.
Necessity of categories comes from the empirical and transcendental use of faculties. Since categories designate the rules for imagination and sensibility needs imagination in order for appearances to be reproduced as images. Without categories experience cannot be possible.

What do you think?

Wednesday, November 25, 2009

Objective Validity

I do not think Kant ever gives a proof or a legitimization for objective validity of categories. He says he will, he does, he did. But it's nowhere to be seen.

I wish modern logic existed when he was writing the Critique! It would be such a different book.

Monday, August 17, 2009

No activity vol.2

I started to read Smiths' review, but really could not give my attention to it. I hope the idea that taking the German test earlier was a good one. Cos' I am so tired. I have to do so much extra work. Two more weeks.

I took a prep. exam and it was SO hard for my level. Oh well, we will see.. My scholarship would not pay for an extension anyway, so I must study without complainin and that's it...

Friday, August 14, 2009


So, I have printed out Peter Smiths overview/blogging on Parsons' book. First I will read that, while reading overviews are usually easier than reading the original text. I have decided that I have been in a break so long from my readings, I am loosing my ability to mathematically philosophize. LOL :) Hence, read a little bit everyday. German is taking over my brain...

Anyway, Simith's review can be found here: