We are tying to find out the meaning of transcendental idealism. We have discovered that it is better not to understand it as the unification (and how could there be such a unification of opposite poles that cancel one another out?) of rationalism and empiricism, but the prising of them apart so as to find, so to speak, hidden beneath this opposition, which appears to cover every possibility, a third alternative. This third possibility is what Kant calls transcendental idealism. One of the routes into what this might mean is rather convoluted expression a priori synthetic judgement.
One of these terms were already in use before Kant, and which he uses in the traditional way, and that is a priori, which was distinguished from a posteriori. An a priorijudgement was one that was necessary and universal, and thus logical, whereas an a posteriori judgement was contingent and grounded in an appeal to experience. Kant, however, invented the term, ‘synthetic’, and here we have to introduce the notion of a predicate proposition. When we say something about something, we predicate something to it. Let us use Kant’s example of ‘all bodies have extension.’ The first concept is the subject of the judgement, ‘body’ and the second the predicate of the subject. Now this judgement for Kant is analytic; that is to say, the predicate can be discovered in the subject simply by analysing it. If I understand the meaning of the concept ‘body’, I know that extension is contained in it. The other kind of judgement is synthetic. Again to use an example of Kant, the following judgement is synthetic: ‘all bodies have attraction’. It is synthetic because under the general description of the ‘body’ it adds a predicate that is not to be found by simply analysing the first concept. No matter what I understand by the term ‘body’. I cannot simply deduce the concept ‘attraction’. This seems clear and simply enough, but where Kant diverges from classical philosophy is in the relation between these new terms and the old ones of a prioriand a posteriori. We would think all analytic judgements would be a prioriand all synthetic ones a posteriori. Yet this is precisely what Kant’s claim to have discovered a priorisynthetic judgements denies. Such judgements could neither be empirical, for they are a priori, nor merely logical, for there are synthetic; that is to say, neither rationalism nor empiricism can explain them. The clue for their existence, for Kant, is to be found in the separation of human knowledge into two faculties, intuition (or sensibility) and the understanding.
The two building blocks of human knowledge for Kant are concepts and intuitions. An intuition is the immediate appearance of the object – the table that you see before your eyes, and the concept is idea or meaning through which you think the object, ‘table’ in general. Although in our mind, we can easily abstract these two sides of human knowledge, in the absence of either nothing truly can be known, for without intuitions concepts would be about nothing, and without concepts intuitions would be chaotic and orderless. To use Kant’s famous saying, concepts without intuitions are empty and, intuitions without concepts are blind:
Without sensibility no object would be given to us, without understanding no object would be thought. Thoughts without content are empty, intuitions without concept are blind. It is therefore just as necessary to make our concepts sensible, that is, to add the object to them in intuition, as to make our intuitions intelligible, that is, to bring them under concepts. These two powers or capacities cannot exchange their functions. The understanding can intuit nothing, the senses can think nothing. Only through their union can knowledge arise. [A51/B79]
Corresponding to intuitions and concepts, there are two faculties of human knowledge: to intuitions, the faculty of sensibility, and to concepts, the faculty of the understanding. For Kant, it is absolutely important, in understanding the kind of beings that we are, that we can only know something through the faculty of sensation; that is to say, that the object must be given to us. There might be other beings (and Kant is thinking of God here) that are able to know things directly through thought (what he calls ‘intellectual intuition’), but we cannot do so. This is a very important limit on the validity of human knowledge. We can think many ideas, but the only ones that have any validity are those that are limited to the sphere of experience, for it is only the conjunction or union of intuitions and concepts that produces legitimate knowledge. Kant’s critique of many of the pseudo problems of philosophy has to do with the illegitimate use of concepts beyond experience.
In the introduction, Kant does not prove the validity of a priori synthetic knowledge, but argues that without out it mathematics, physics and metaphysics would not be possible. The argument, if you like, works, backwards. We all accept that mathematics and the others forms of knowledge exist, and that they must be a priori and synthetic, therefore the task of philosophy is to prove their validity. However, some might argue that it is not even the case that these discourses are a priori and synthetic. We might argue for example that mathematics is a priori and analytic and physics is a posteriori and synthetic.
Let us then take the case of mathematics. which Kant spends sometime describing in the introduction. Why does Kant argue that it is synthetic and a priori and not analytic? To answer this we need to look a little closer at the difference between the analytic and synthetic that we have already introduced. First of all how can we recognise a priori judgements? Kant’s response is that every a priori judgement fulfils two criteria: universality and necessity. Thus an a posteriori judgement could never be necessary or universal, since it could always be contradicted by a future experience. As we have already said, we tend to think that the difference between a priori and a posteriori statement is the same as the difference between analytic and synthetic statement. Thus every analytic statement is a priori and every synthetic statement is a posteriori. So following this reasoning, since mathematics statements are clearly a priori, since they fulfil the criteria of universality and necessity, then they must be analytic. Yet it is precisely this designation that Kant disputes.
Kant defines analytic in terms of the relation between the subject and the predicate. An analytic statement is one in which the predicate is contained in the subject. So in the statement, which we have already used, ‘all bodies are extended’, the subject ‘body’ contains necessarily and universally the predicate ‘extended’. It is important to be precise here; otherwise why Kant thinks mathematical statements are a priori but synthetic will be obscure to us. What does it mean to say that one concept is contained in another? This a relation between a higher genus and lower species. The genus is contained in the species and the species are contained under the genus. Thus under the concept ‘metal’ is contained the concepts ‘gold’ or ‘copper’. A concept’s content is general concepts contained in it, and its logical extensions are the specific concepts under it. We can understand this idea of containment in terms of logical division. Thus a concept is logical divided in terms of its extensions. We can divide the concept number therefore into odd and even numbers. This division is logical because it follows the rules of completeness and exclusivity. So the predicate ‘odd’ covers completely the genus ‘number’ and no odd number can be even. This means that conceptual relations are reciprocal. Thus every extension of the concept A contains A as part of its content. A lower concept must contain whatever a higher concept contains, and a visa versa. Without this concept containment, then we are not facing an analytic judgement. This has nothing to with the idiosyncratic definitions of an individual but how one concept is contained in another.
When it comes to the introduction of the Critique, the key question for Kant is whether mathematical statements follow analytic containment. It is clear that mathematical judgements are a priori because they are universal and necessary. But are they analytic? The problem is that when Kant declares that they not and our synthetic, he seems merely to state it rather than prove it.
One might initially think that […] ‘7+5=12’ is a merely analytic proposition […]. Yet if one considers it more closely, one finds that the concept of the sum of 7and5 contains nothing more than the unification of both numbers in a single one, through which it is not at all thought what this single number is […]. The concept of twelve is by no means already thought merely by my thinking that unification of seven and five, and no matter how long I analyse my concept of such a possible sum, I will still not find twelve in it. [B15]
What Kant has to show is that there is not a containment relation between the concept ‘7 + 5’ and the concept 12. The argument would be that mathematical relations between numbers cannot be expressed in containment relations. Remember that containment relations are reciprocal. If A contains B, then B must contain A, and whatever A includes or excludes, B must include or exclude. This is precisely what does not hold with mathematical expressions. If 12 is contained in the concept ‘7+5’ then it must include and exclude what it does, but this would be mean that that concept ‘7+5’ would exclude ‘7’ and ‘5’, since 12 ≠ 7, nor 12 ≠ 5. What is not contained in the concepts ‘7’ and ‘5’, therefore is the sum concept ‘7 + 5’. Something else is added there, which is the relation between these three numbers. This is clearer if we look at the following proposition, which Kant used in letter to Johann Schultz: ‘3 + 5’ = ‘2 × 4’. Here no-one would think that the two concepts have the same content, since they involve different numbers with different operations even though they might refer to the same object (the number ‘8’).
It would appear then that mathematics proves the existence of a priori synthetic statement. The question for the rest of the Critique is why do these judgements exist in mathematics, physics and mathematics, and do we have the right to use them? What we will discover is that origin of mathematics is in what Kant calls pure intuition and the pure understanding, and not in logic or experience.
Allison, H.E., 2004. Kant’s Transcendental Idealism, New Haven: Yale University Press.
Guyer, P. ed., 2010. The Cambridge Companion to Kant’s Critique of Pure Reason, Cambridge; New York: Cambridge University Press.
 The first example is taken from The Critique of Pure Reason, whereas the second is from Kant’s lectures on logic, which is referred to by Allison (2004, p.76).
 This is separation of philosophy; in experience they always go together. Thus I always see something as something, and never just see it.
 We need to underline here that intuition for Kant does not mean the power of the mind by which it immediately perceives the truth of things without reasoning or analysis; rather it is immediate sensation. These immediate sensations, however, are still representation. I don’t see a ‘red sensation’. I see ‘red’.
 An excellent account of concept containment can be found in Anderson’s chapter in The Cambridge Companion to Kant’s Critique of Pure Reason (Guyer 2010, pp.73–92), which our explanation follows pretty much to the letter.