So the only one of these that jumps out at me as being really unhelpful is
“Kant and Strawson on the Content of Geometrical Concepts.” This paper considers Kant’s understanding of conceptual representation in light of his view of geometry. [...] While conceding that Kant confuses pure and applied geometry, P. F. Strawson tries to preserve the interest of his view. Strawson seeks to explain how the application of geometry can be independent of experience. [...] I sketch a way of reconciling Strawson’s interpretation of “pure intuition” (on which it represents objects as we imagine, or are prepared to picture, them) with Kant’s view that it proves the applicability of concepts independently of experience. Pure intuition can be taken, in the spirit of Strawson’s interpretation, to represent procedures for constructing objects that fall under the concepts. I argue that on Kant’s view, the representation of such procedures indeed yields a priori knowledge of the applicability of concepts.
This fails at multiple levels. First it fails, because pretty much everything Kant wrote about geometry runs into the serious problem that his whole idea is deeply connected to Euclidean geometry being the one, true correct geometry. Second, this runs into the earlier discussed problem of trying to discuss what major philosophers meant, as if that had intrinsic interest. Third, a glance strongly suggests that they are ignoring the large body of actual developmental psych data about how children actually do and do not demonstrate intuitions for their surrounding geometry.
I don’t know enough about the subjects to say much about the Skow, Uzquiano, and Button although I suspect that the third is confusing linguistic with metaphysical issues.
I agree with most of your objections and I think we must, at this point, notice how different this selection of articles looks from what Luke originally presented.
Since you’re criticizing an article based on my own chosen excerpts of it, it would be irresponsible of me not to give fuller quotes so that Dunlop can respond:
Subsequent advances in mathematics and physics appeared to discredit Kant’s view of intuition. They showed that no geometry can be known a priori to apply to physical space. In their light, Kant’s view appeared to rest on a confusion between pure geometry, which is known a priori but empty, and its applications. While conceding that Kant confuses pure and applied geometry, P. F. Strawson tries to preserve the interest of his view. Strawson seeks to explain how the application of geometry can be independent of experience. Kant holds that the applicability of geometrical concepts to all objects represented in empirical intuition (ordinary sense-perception) is proved by our ability to represent objects falling under them in pure intuition. Strawson interprets Kant’s “pure intuition” as the capacity to give ourselves “pictures” in imagination. He takes Kant to argue that because all use of geometrical concepts involves picturing, what holds of all pictures we can give ourselves must hold of all objects represented through sensibility.
A preliminary goal of this paper is to defend Strawson’s view that Kant intends to explain our ability to formulate the criteria (marks) by which we recognize instances of a concept. [...] Strawson is also right that on Kant’s view, concepts acquire content of this kind (criteria of application) through an exercise of the imagination. But Strawson misunderstands the imaginative activity through which concepts are defined. He regards it as an application of concepts, to pictures, which is a priori in the sense that it mediates all use of the concepts in sense-experience. But I argue (in §3) that for Kant, it is a priori in the stronger sense that it is required even to possess the concepts. Kant holds that mathematical concepts are formed by defining them. Since we cannot possess them without grasping definitions, which assure the universal applicability of the concepts, no experience (even that of inspecting pictures) can provide any further guarantee of their applicability. I explain why, according to Kant, experience is not needed either to formulate or to ascribe the marks included in mathematical concepts.
Strawson points us toward a key element of Kant’s theory of geometry, but it does not fit the place he gives it in Kant’s catalogue of kinds of representation. I sketch a way of reconciling Strawson’s interpretation of “pure intuition” (on which it represents objects as we imagine, or are prepared to picture, them) with Kant’s view that it proves the applicability of concepts independently of experience. Pure intuition can be taken, in the spirit of Strawson’s interpretation, to represent procedures for constructing objects that fall under the concepts. I argue that on Kant’s view, the representation of such procedures indeed yields a priori knowledge of the applicability of concepts. But because these procedures must be represented as general, and intuition represents particulars, it would be wrong to understand pure intuition as the representation of these procedures. I explain that on Kant’s view, procedures for constructing objects are represented as “schemata”, which are distinct from concepts and intuitions.
My main objection to Strawson is that he overlooks the implications of Kant’s view of definition. Kant uses the definition of geometrical concepts to illustrate the sensible faculty’s role in cognition. He thinks it is distinctive of mathematical concepts that they are formed by defining them. [...] If Kant’s theory of geometry is defunct, we should join Strawson’s effort to detach from it insights that can claim to last. So to defend the relevance of Kant’s theory (even for interpreting the Critique), I must deal with the objections that lead Strawson to minimize its role. If Kant’s view that mathematical concepts have schemata rested merely on an inability to conceive them more abstractly, he could still be charged with confusing pure and applied geometry. But Kant holds that outside of mathematics, concepts can be formed independently of the constraints imposed by sensibility. Since we can form (but not give schemata to) concepts that do not accord with (Euclidean) geometry, the formation of geometrical concepts involves a kind of choice (as I explain in §6).
Kant’s view of geometry thus has an affinity, overlooked by Strawson, with “conventionalist” views. Yet this choice is not arbitrary for Kant in the way it is on later views. Kant holds that the choice of concepts intended for mathematical use is informed by cognition of their applicability (as I explain in §7). Specifically, we voluntarily restrict the understanding to forming only those concepts whose schemata we can represent (which entails that objects answering to them can be represented in accordance with the conditions on our perception). By choosing to form only applicable concepts, we license ourselves to ascribe their marks independently of experience: in particular, without regard to our experience of how concrete material objects fall short of geometry’s specifications. So our prerogatives to fix criteria for the application of mathematical concepts, and stipulate their satisfaction, ultimately rest on a more fundamental ability to perceptually represent objects that answer to the concepts.
The criticism has been made that philosophers waste too much time on historical exegesis; but I found surprisingly very little historical work in Noûs, and even this Kant stuff is surprisingly relevant to some of the contemporary issues we ourselves have been debating recently—concerning the relationship between imagined or constructed ‘mathematical reality’ and the empirical world, the dependence of logical truth upon thought, etc.
Ok. This excerpt gives me a much higher opinion of the piece in question and substantially reduces the validity of my criticisms. Since this was the article that most strongly seemed to support the sort of point that Luke was making, I’m forced to update strongly against Luke’s selected papers being at all representative.
First it fails, because pretty much everything Kant wrote about geometry runs into the serious problem that his whole idea is deeply connected to Euclidean geometry being the one, true correct geometry.
I don’t recall him ever restricting himself to only Euclidean geometry. In Critique of Pure Reason, “geometry” is mentioned twenty times (each paragraph a separate quote; Markdown is being dumb):
Just as little is any principle of pure geometry analytical. “A straight line between two points is the shortest,” is a synthetical proposition.
Thus, moreover, the principles of geometry—for example, that “in a triangle, two sides together are greater than the third,” are never deduced from general conceptions of line and triangle, but from intuition, and this a priori, with apodeictic certainty.
Geometry is a science which determines the properties of space synthetically, and yet a priori. What, then, must be our representation of space, in order that such a cognition of it may be possible? It must be originally intuition, for from a mere conception, no propositions can be deduced which go out beyond the conception, and yet this happens in geometry.
Thus it is only by means of our explanation that the possibility of geometry, as a synthetical science a priori, becomes comprehensible.
Take, for example, the proposition: “Two straight lines cannot enclose a space, and with these alone no figure is possible,” and try to deduce it from the conception of a straight line and the number two; or take the proposition: “It is possible to construct a figure with three straight lines,” and endeavour, in like manner, to deduce it from the mere conception of a straight line and the number three. All your endeavours are in vain, and you find yourself forced to have recourse to intuition, as, in fact, geometry always does.
Geometry, nevertheless, advances steadily and securely in the province of pure a priori cognitions, without needing to ask from philosophy any certificate as to the pure and legitimate origin of its fundamental conception of space.
Footnote: Motion of an object in space does not belong to a pure science, consequently not to geometry; because, that a thing is movable cannot be known a priori, but only from experience.
On the other hand, the self-evident propositions as to the relation of numbers, are certainly synthetical but not universal, like those of geometry, and for this reason cannot be called axioms, but numerical formulae.
Empirical intuition is possible only through pure intuition (of space and time); consequently, what geometry affirms of the latter, is indisputably valid of the former.
But in this case, no a priori synthetical cognition of them could be possible, consequently not through pure conceptions of space and the science which determines these conceptions, that is to say, geometry, would itself be impossible.
But mathematics does not confine itself to the construction of quantities (quanta), as in the case of geometry; it occupies itself with pure quantity also (quantitas), as in the case of algebra, where complete abstraction is made of the properties of the object indicated by the conception of quantity.
Thus, when one quantity is to be divided by another, the signs which denote both are placed in the form peculiar to the operation of division; and thus algebra, by means of a symbolical construction of quantity, just as geometry, with its ostensive or geometrical construction (a construction of the objects themselves), arrives at results which discursive cognition cannot hope to reach by the aid of mere conceptions.
We shall, accordingly, show that the mathematical method is unattended in the sphere of philosophy by the least advantage—except, perhaps, that it more plainly exhibits its own inadequacy—that geometry and philosophy are two quite different things, although they go band in hand in hand in the field of natural science, and, consequently, that the procedure of the one can never be imitated by the other.
Of the two kinds of a priori synthetical propositions above mentioned, only those which are employed in philosophy can, according to the general mode of speech, bear this name; those of arithmetic or geometry would not be rightly so denominated.
For the assertion that the reality of such ideas is probable is as absurd as a proof of the probability of a proposition in geometry.
Other than this last quote (which is simply wrong), all of the other mentions consider geometry either as 1) a mere example or 2) in the context of phenomenal experience, which is predominately Euclidean for standard human beings on Earth. One could easily take it as a partial statement of the psychological unity of humankind.
He doesn’t discuss it that much, but there’s a strong argument that it is operating the background 1 (pdf). The same author as linked wrote an essay about this, but I can’t find it right now.
This is strange, because your link is about Kant disagreeing with other philosophers on the nature of Euclid’s parallel postulate. I took your claim to be that because Kant was seemingly only aware of Euclidean geometry, he used properties specific to only Euclidean geometry in his discussion of geometry.
Show me explicitly where this “operating in the background” is, and I’d be more convinced.
Hmm, ok. Rereading the link and thinking about this more, it looks like I’m either strongly misremembering what it said or am just hopelessly confused. I’ll need to think about this more.
First it fails, because pretty much everything Kant wrote about geometry runs into the serious problem that his whole idea is deeply connected to Euclidean geometry being the one, true correct geometry.
That’s true, but it doesn’t sound relevant to the subject of the article.
Second, this runs into the earlier discussed problem of trying to discuss what major philosophers meant, as if that had intrinsic interest.
A solid blow.
Third, a glance strongly suggests that they are ignoring the large body of actual developmental psych data about how children actually do and do not demonstrate intuitions for their surrounding geometry.
That might be relevant to Strawson’s view, I’m not actually sure what he says, but it’s not relevant to Kant’s view. ‘A priori’ does not mean ‘innate’ or biologically determined.
That might be relevant to Strawson’s view, I’m not actually sure what he says, but it’s not relevant to Kant’s view. ‘A priori’ does not mean ‘innate’ or biologically determined.
It doesn’t to mondern philosophers, but the way it was used by Kant it seems like he meant it very close to how we would use “innate”.
No, Kant thought that you could only have synthetic a priori knowledge if you already had a fair amount of experience with the world. Synthetic a priori knowledge is knowledge which rests on experience (Kant thinks all knowledge begins with experience), but it doesn’t make reference to specific experiences. Likewise, analytic a priori knowledge requires knowledge of language and logic, which, of course, is not innate either. Kant doesn’t think there’s any such thing as innate knowledge, if this means knowledge temporally prior to any experience.
So the only one of these that jumps out at me as being really unhelpful is
This fails at multiple levels. First it fails, because pretty much everything Kant wrote about geometry runs into the serious problem that his whole idea is deeply connected to Euclidean geometry being the one, true correct geometry. Second, this runs into the earlier discussed problem of trying to discuss what major philosophers meant, as if that had intrinsic interest. Third, a glance strongly suggests that they are ignoring the large body of actual developmental psych data about how children actually do and do not demonstrate intuitions for their surrounding geometry.
I don’t know enough about the subjects to say much about the Skow, Uzquiano, and Button although I suspect that the third is confusing linguistic with metaphysical issues.
I agree with most of your objections and I think we must, at this point, notice how different this selection of articles looks from what Luke originally presented.
Since you’re criticizing an article based on my own chosen excerpts of it, it would be irresponsible of me not to give fuller quotes so that Dunlop can respond:
The criticism has been made that philosophers waste too much time on historical exegesis; but I found surprisingly very little historical work in Noûs, and even this Kant stuff is surprisingly relevant to some of the contemporary issues we ourselves have been debating recently—concerning the relationship between imagined or constructed ‘mathematical reality’ and the empirical world, the dependence of logical truth upon thought, etc.
Ok. This excerpt gives me a much higher opinion of the piece in question and substantially reduces the validity of my criticisms. Since this was the article that most strongly seemed to support the sort of point that Luke was making, I’m forced to update strongly against Luke’s selected papers being at all representative.
I don’t recall him ever restricting himself to only Euclidean geometry. In Critique of Pure Reason, “geometry” is mentioned twenty times (each paragraph a separate quote; Markdown is being dumb):
Other than this last quote (which is simply wrong), all of the other mentions consider geometry either as 1) a mere example or 2) in the context of phenomenal experience, which is predominately Euclidean for standard human beings on Earth. One could easily take it as a partial statement of the psychological unity of humankind.
He doesn’t discuss it that much, but there’s a strong argument that it is operating the background 1 (pdf). The same author as linked wrote an essay about this, but I can’t find it right now.
This is strange, because your link is about Kant disagreeing with other philosophers on the nature of Euclid’s parallel postulate. I took your claim to be that because Kant was seemingly only aware of Euclidean geometry, he used properties specific to only Euclidean geometry in his discussion of geometry.
Show me explicitly where this “operating in the background” is, and I’d be more convinced.
Hmm, ok. Rereading the link and thinking about this more, it looks like I’m either strongly misremembering what it said or am just hopelessly confused. I’ll need to think about this more.
Thinking about this less and something else more is also a good option.
That’s true, but it doesn’t sound relevant to the subject of the article.
A solid blow.
That might be relevant to Strawson’s view, I’m not actually sure what he says, but it’s not relevant to Kant’s view. ‘A priori’ does not mean ‘innate’ or biologically determined.
It doesn’t to mondern philosophers, but the way it was used by Kant it seems like he meant it very close to how we would use “innate”.
No, Kant thought that you could only have synthetic a priori knowledge if you already had a fair amount of experience with the world. Synthetic a priori knowledge is knowledge which rests on experience (Kant thinks all knowledge begins with experience), but it doesn’t make reference to specific experiences. Likewise, analytic a priori knowledge requires knowledge of language and logic, which, of course, is not innate either. Kant doesn’t think there’s any such thing as innate knowledge, if this means knowledge temporally prior to any experience.
This has it about right: http://en.wikipedia.org/wiki/A_priori_and_a_posteriori#Immanuel_Kant