24.01 Classics of Western Philosophy
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1 1 Plan: Kant Lecture #2: How are pure mathematics and pure natural science possible? 1. Review: Problem of Metaphysics 2. Kantian Commitments 3. Pure Mathematics 4. Transcendental Idealism 5. Pure Natural Science 2 Review/Set Up: On Tuesday Prof. Hagen introduced Kant s project of addressing the problem of metaphysics. Metaphysics is all messed up: Does not possess universal validity ( universal and lasting recognition ). This is not meant only as an external approval of the public but as an internal consensus of the scientific community. Does not show results and progress ( moving around the same spot without gaining a single step ). In terms of advancement Metaphysics is still where it was in the days of Aristotle. Does not have criteria to distinguish sound knowledge from mere ignorance. This is the most severe defect. It accounts for the fact that in Metaphysics everyone can pass judgments with authority, which means that there is no real expertise in matters of metaphysical knowledge. Kant addresses this problem in the Prolegomena by demonstrating how metaphysical judgments are possible. This at least gets us that metaphysics is possible! Metaphysical judgments are synthetic a priori, substantive yet necessary. Prolegomena becomes: how are synthetic a priori judgments possible? So the question of the He uses the legit and well-established sciences of pure mathematics and pure natural science to help with the case of metaphysics. Both of these sciences involve synthetic a priori judgments, so if we can explain how those are possible we will have a model for how to deal with metaphysical judgments as well. The goal for today is to investigate how Kant explains these other legit sciences. 3 Kantian Commitments. As Prof. Hagen said, reading Kant involves learning a new language and to make sense of his philosophy we need to understand and adopt some important distinctions. We are already familiar with analytic/synthetic and a priori/a posteriori. The most important distinction that will be playing a role in our discussion today is between intuitions and concepts: 1 of 6
2 Kant believes that all of our knowledge, those objective cognitions, is comprised of two distinct representations: intuitions and concepts. Quoting from the CPR: An intuition refer directly to the object and is singular; a concept refers to the object indirectly, by means of a characteristic that may be common to several things. (A320/B376-7) Intuitions are singular, immediate representations. Concepts are general, mediate representations. So THAT vs. chair Note that we need concepts to turn our representations into communicable judgments. To say That chair is green. we need the concepts chair and green. Further, Kant says that these two kinds of representations are governed by particular mental faculties. The faculty of sensibility governs our intuitions and the faculty of understanding governs our concepts. Exactly what I mean by governs here will come out more as we talk about how pure mathematics is possible. Which we now turn to. 4 How is pure mathematics possible? Before going into the details of how pure mathematical judgments are possible, it s important to note that the general question of the Prolegomena was really shaped by his conception of mathematics and its achievements as well-grounded science. Really, his thoughts on mathematics are a crucial and central component of his critical philosophical system. And as crazy as they may seem, these thoughts continue to inform important questions in the metaphysics and epistemology of mathematics to this day! Now, onto the question: how are pure mathematical judgments possible? Using the distinctions we just discussed, Kant claims that mathematical judgmentsare synthetic, a priori and are always intuitive, and are thus always represented through the mental faculty of sensibility. How are intuitive synthetic a priori judgments possible? It s easy to see why Kant thinks that pure mathematics should be considered a priori; they are necessary and so cannot be derived from experience. Why, though, does he think that these judgments are synthetic? He claims that mathematical judgments synthesize a subject and predicate concept rather than merely explicate or analyze a subject concept into its constituent logical parts. Let s walk through an example of how Kant sees this working: = 12 He first claims (negatively) that no matter how long he analyzes the concept of a possible sum of 5 and 7 that he will never find 12 in it. Remember the containment metaphor of analytic/synthetic judgments. Then he argues (positively) that one might go beyond the concepts of five and seven, seeking assistance in the intuition that corresponds to one of the two, one s five fingers, say... and one after another add the units of five given in the intuition to the concept of seven... and thus see the number 12 arise (B15). Thus, he takes it to follow that the necessary truth of an arithmetic proposition cannot be established by any method of logical or conceptual analysis, but can be established by intuitive synthesis. He also demonstrates this with a geometrical example in the text. 2 of 6
3 The key, then, is that the synthesis doesn t happen via concepts but, rather via our intuitions. This is what distinguishes mathematical judgments from philosophical ones: they are always intuitive. But then we run into another difficult question: how are a priori intuitions possible? Recall that intuitions are singular, immediate perceptions; it seems like these things could only be given to us by experience. If pure mathematical judgments are possible, Kant needs to explain this. His answer is that we can intuit via the pure forms of sensibility: He addresses this in 9: Therefore in one way only can my intuition anticipate the actuality of the object, and be a cognition a priori, viz., if my intuition contains nothing but the form of sensibility, which in me as subject precedes all the actual impressions through which I am affected. (282) And also in 10: Accordingly, it is only the form of sensuous intuition by which we can intuit things a priori, but by which we can know objects only as they appear to us (to our senses), not as they are in themselves; and this assumption is absolutely necessary if synthetic propositions a priori be granted as possible or if, in case they actually occur, their possibility is to be determined beforehand. (283) When discussing the different mental faculties above I said that the faculty of sensibility governs our intuitions and now we can make more sense of what this means. This faculty imposes itself onto all of our representations of the world via these pure forms of sensibility. It turns out that the pure forms of sensibility are those of space and time. All of our intuitions are formed by space and time; they underlie all of our intuition of the world. Thus, importantly, space and time are not properties of things in themselves. Filter analogy: we can think of things in themselves out there, doing their thing. Because we are the kind of beings that we are (we have certain mental faculties, one of which is the sensibly) when we observe the world all of our representations of the world are put through a filter of space and time. Phenomena/Noumena: We can only have knowledge of phenomena, the objects of our experience. We can never have knowledge of things in themselves, the noumena, because things must filter through our mental faculties in order for us to know anything about them. Interesting to think way back to when we read Plato s Phadeo and his thoughts on the Forms vs. particulars. QUESTION: Does someone remember what Plato thought about the forms and particulars as objects of knowledge? QUESTION: How does this differ with Kant s view? We ve made it all the way to way Kant s Copernican Revolution. This is one of the main takeaways from Kant s philosophical system: Thus far it has been assumed that all our cognition must conform to objects. On that presupposition, however, all our attempts to establish something about them a priori, by means of concepts through which our cognition would be expanded, have come to nothing. Let us, therefore, try to find out whether we shall not make better progress in the problems of metaphysics if we assume that objects must conform to our cognition. (CPR, Bxvi) Given that doing metaphysics from one position has left us with an unsatisfactory pseudo-science, Kant suggests we change our position and go from there. Of course this isn t an argument for taking this view but rather a suggestion for making more progress. 3 of 6
4 Okay, so we just walked through a lot of complicated stuff. Let s think now about how this helps us address the question of how pure mathematics are possible. The takeaway is that we can have intuitions a priori via the pure forms of sensibility. They don t require any experiential input, we can see them in our minds eye, in a sense. And they are necessary because they hold in this purest sense. Thus, Kant concludes in 11: The problem of the present section is therefore solved. Pure mathematics, as synthetic cognition a priori, is possible only by referring to no other objects than those of the senses. At the basis of their empirical intuition lies a pure intuition (of space and time) which is a priori. I will leave more detailed discussions of how Kant explains geometry and arithmetic via pure sensibility for discussion in recitation. 5 Transcendental Idealism. Okay, we have just explained how pure mathematics is possible. The next question is how pure natural science is possible. But before we solve that problem I want to discuss Kant s philosophy of Transcendental Idealism, which is the view we just discussed regarding the phenomena/noumena distinction. This is Kant s big thing, his Copernican Revolution. In the First Part of the Transcendental Question, in the Remarks, he address some potential problems for the view and I think it is helpful if we consider them briefly. The first potential problem is that since mathematics is now only true of the phenomenal realm how can we think of mathematical judgments as objective? Kant says why doesn t it follow that the propositions of geometry are not determinations of a mere creation of our poetic imagination, which could therefore not be referred with assurance to actual objects? (287) How does Kant get out of this potential problem? Mathematical truths are then objectively true because they hold for all phenomena, and we can only know about phenomena:... they are necessarily valid of space, and consequently of all that may be found in space, because space is nothing but the form of all external appearances, and it is this form alone in which objects of the sense can be given to us. (287) The second potential problem is that Kant s theory could be taken as idealism. First, what s idealism? Let s think back to Descartes dualist metaphysics. He thought that at the fundamental metaphysical level there are two kinds of things: mind and body. Then we looked at Cavendish s physicalism; she argues that there is only ONE kind of fundamental metaphysical stuff: body. Idealism is sort of the opposite of physicalism; it is the view that there is only ONE kind of fundamental metaphysical stuff, mind. We haven t looked at any idealist views in this course, but a very famous one is given by Berkeley. We can see why one might think Kant s view is idealistic: the only things we can know about are those that exist via our mental faculties. Why does Kant claim that his view is not idealism? 4 of 6
5 I say that things as objects of our senses existing outside us are given, but we know nothing of what they may be in themselves, knowing only their appearances, i.e., the representations which they cause in us by affecting our senses... The existence of the thing that appears is thereby not destroyed, as in genuine idealism, but it is only shown that we cannot possibly know it by the senses as it is in itself. (289) 6 How is pure natural science possible? Okay, let s recap where we are and what we ve worked through. Kant wants to show how metaphysical synthetic a priori judgments are possible. To do this he first shows how mathematical judgments, which are synthetic a priori are possible. To provide this explanation Kant has to introduce his hefty view of Transcendental Idealism which states that we can only know about the phenomenal world. This solves the problem for pure mathematics because it gives us a way to intuit things a priori. Now we move to pure natural science. Given that we can never know about things in themselves, how are universal scientific laws possible? How can we say that there are objective laws like substance is permanent and every event is determined by a cause? We want to remember that Kant is assuming that everything is all well and good with pure natural science; we are just asking how these judgments are possible. The first important step is to acknowledge that nature consists of all and only objects of experience, not things in themselves. Thus, pure natural science can only give us universal laws for the phenomenal realm. The second step involves bringing in the other mental faculty that I discussed at the beginning of class: the understanding, which governs our conceptual representations. Just as there are pure forms of sensibility with govern all intuitions, there are pure concepts of the understanding which govern all concepts. He makes another important distinction between two kinds of empirical judgments: judgments of experience and judgments of perception: Judgments of experience always require, besides the representation of the sensuous intuition, special concepts originally generated in the understanding, which make the judgment of experience objectively valid. (298) Judgments of perception require no such concept. All judgments start off as judgments of perception and then, after we give them a new reference (outside of just ourselves), they become judgments of experience, which signify necessary universality validity. Just as all representations get filtered through our faculty of sensibility, in order to become a judgment of experience our intuitions must be filtered through our faculty of understanding: Middle of first paragraph of 20: Quite another judgment therefore is required before perception can become experience. The given intuition must be subsumed under a concept which determines the form or judging in general with regard to the intuition, connects the empirical consciousness of the intuition in conscious in general, and thereby procures universal validity for empirical judgments. A concept of this nature is a pure a priori concept of the understanding, which does nothing but determine for an intuition the general way in which it can be used for judging. (300) These pure concepts of the understanding underlie all of our judgments of experience, just as the pure intuitions of sensibility underlie all of our intuitions. 5 of 6
6 How does this help us with pure natural science? Because all objects of knowledge must be filtered through these pure concepts of the understanding they necessarily hold for all of those objects of experience and, thus, we can consider them to be universal natural laws. One thing to think about in recitation tomorrow is how Kant uses the pure concepts of the understanding to address Hume s challenge that we discussed last week. 7 Conclusions. Let s think back to our discussion of rationalism and empiricism before we leave today. The rationalists claim that some of our knowledge, the first principles as Descartes calls it, are internal to us, we know them innately. The empiricists claim that all of our knowledge comes to us through experience. Kant is often touted as bringing together rationalism and empiricism, in showing how all of our knowledge requires both innate elements and our experience. Think about why he is thought to have solved this problem? Perhaps we can all discuss this in recitation tomorrow... Next week we will find out how metaphysics is possible! Please read through selections for Tuesday s lecture. 6 of 6
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