Critical Assessment of Geometry as a Synthetic System
Eighteenth century German philosopher Immanuel Kant establishes in his Critique of Pure Reason that there are certain cognitions of the mind that are independent from experience and sense impressions, calling them a priori cognitions. Of all the a priori concepts Kant exposits in his Critique, this paper focus on the first and foremost that was examined; namely, the concept of space. To derive space necessarily independent of experience, Kant relies on geometry as "a science that determines that properties of space synthetically and yet a priori" (Kant B40). However, a later philosopher A. J. Ayer objects the Kantian notion of the synthetic a priori; criticizing that geometry is in fact analytic and tautological. In this paper, I will define the distinction between the analytic and synthetic propositions (in both Kant and Ayer's point of view), and assess both views of the a priori critically with objections to both sides. Finally, I will defend Ayer's argument, that geometry cannot be synthetic and is instead analytic, concluding that it is a stronger argument than that of Kant's.
It is true that all of our cognitions begin with experience, as our knowledge of the world is initially taught to us by others. However, Kant recognizes that simply because cognition begins with experience does not mean it must necessarily arise with experience; there are certain cognitions that can be justified absolutely independently of experience. These cognitions are a priori, "they are distinguished from empirical cognitions, whose sources are a posteriori, namely, in experience" (Kant B2). Within the category of a priori knowledge, Kant makes the distinction between analytic propositions, and synthetic propositions. The definitions are offered as a relationship between a subject, the phrasal constituent which we are cognizing, and the predicate, the phrasal constituent that modifies the subject. Kant believes that when "the predicate B belongs to the subject A as something that is covertly contained in this concept A" (Kant A6/B10), the proposition is analytic. Likewise, if the predicate "B, though connected with concept A, lies quite outside it" (Kant A6/B10-A7), the proposition is synthetic. Kant gives an exclusive explanation to the particular concept of the synthetic a priori, as he relies heavily on it to further develop his argument. He gives the proposition "everything that happens has its cause" as an example. This proposition is necessarily synthetic because "the concept of a cause lies quite outside that earlier concept and indicates something different from what happens; hence it is not part of what is contained in this latter presentation" (Kant A9/B13). At the same time, it is also a priori because when discovering the predicate, Kant expresses that our minds rely on an "unknown" that cannot be experience, since it adds cause to an event with "greater universality than experience can provide, but also with the necessity's being expressed" (Kant A9/B13).
The clear distinction between analytic and synthetic propositions is important to Kant, because the property of space, the form of intuition of our outer sense, is derived from the synthetic a priori cognition of geometry. To express the synthetic a priori properties of geometry, Kant raises the example of a "straight line between two points is the shortest [distance]" (Kant B16). This proposition is a priori because the concept expressed by the proposition is necessary without contradiction, which is a property experience can ever achieve. More importantly, Kant shows the synthetic properties of the proposition by arguing that the notion of "straight" is a quality that tells him nothing of the magnitude of the line. The notion of being the "shortest" is a predicate that adds to the subject of the "straight line between two points", also indicating that the predicate is not contained within the concept, and cannot be extracted from the subject. To have a proper understanding of the proposition, we must then synthesize the predicate with the subject using our intuition. Hence, the given geometrical proposition is entirely synthetic (Kant B16). Likewise, as Kant presupposes in this argument, because most geometrical propositions are given similar to the example examined, it is justifiable to establish the concept of geometry as synthetically a priori.
Kant naturally thinks that geometry is the foundation of space because it is certain and necessary, but as contemporary empiricist A.J. Ayer points out, if we do not question Kant's definition of the properties of geometry, "then we may be inclined to accept [his] hypothesis that space is
the only possible explanation of our a priori knowledge of these synthetic propositions" (Ayer 82) Ayer criticizes Kant for making an invalid leap from the concept of geometry to the concept of space, as geometry in itself is not about anything at all, let along about physical space. Geometry consists of axioms and theorems, as Ayer points out, and the "axioms of geometry are simply definitions, and that the theorems of geometry are simply the logical consequences of these definitions" (Ayer 82). Axioms are analytic propositions that do not add to anything because there is nothing from which axioms are derived from. Therefore, if a set of axioms are given to a physical presentation of geometry, Ayer states that "we can proceed to apply the theorems to the objects which satisfy the axioms" (Ayer 82) Examining whether or not geometry applies to the physical world is not a question of geometry in itself, but rather, an empirical one. This is so because a proposition that tells of the application of geometry in the physical world is intrinsically different from a geometrical proposition as such. The former lies outside of the system of geometry, because "all that geometry itself tells us is that if anything can be brought under the definitions, it will also satisfy the theorems" (Ayer, 83). This distinction is to prove that geometry in itself is in fact a purely logical system consisting of purely analytic propositions, which are not based on synthesis of intuitions as Kant had earlier concluded.
While it is true that in one aspect, geometry can act simply as a set of theorems that can be logically derived from analytical axioms, but for the system itself to be purely analytical, Ayer must account for the synthetic component geometry is built upon. To arrive at geometrical theorems, or any other proposition that can be derived in geometry, two constituent factors are involved: geometrical axioms and mathematics. Because geometry is a product of axioms and mathematics, for it to be necessarily analytic means that both the axioms and mathematics must also be analytic. Axioms are, as previously established, analytic propositions, because they reveal no additional information to any matter of fact. Mathematics, on the other hand, is a synthetic system.
Kant sets out to demonstrate that mathematical propositions are synthetic by using the simple equation "7 + 5 = 12". First, he indicates that "7 + 5" "contains nothing more than the union of the two numbers into one; but [by] thinking [of] that union we are not thinking in any way at all what that single number is that unites the two" (Kant B15). That is to say, when I cognize of the coming together of quantities "7" and "5", I do not automatically think of the quantity "12". Because the concept "12" is not naturally contained within the concept of a union between "7" and "5", I cannot find within the concept a notion of "12". It is necessary for me to conform one of the numbers to my intuition, for example, "5", and gradually add it up to the quantity of "7". It is only after this process can I come to conclude that the union of "7" and "5" is "12". To better prove this hypothesis, Kant also states that this synthetic distinction becomes much more observable when dealing with large numbers, for our mind is too familiar with small numbers (such as seven and five) to make out the distinction clearly (Kant B16). For example, if I were given the proposition "91 x 79 = 7189", I would not be able to think of "7189" as I think of the product of "91" and "79", as it is not contained in the concept of "91 x 79". It is only after I have gone through the process of multiplication can I intuit that the predicate "7189" is in fact the product of the previous two numbers, which would consequently add to the subject of "91 x 79". From this, Kant concludes that "arithmetic propositions are therefore always synthetic" (Kant B16).
To better examine the criticism Ayer has raised against Kant's idea of the synthetic system of mathematics, we must first address the issues Ayer pointed out in regards to Kant's definition of the analytic and synthetic propositions. Ayer suggests that to distinguish between the analytic and the synthetic, Kant relies on two different criteria that are not at all the same (Ayer 78). To determine that the proposition "7 + 5 = 12" is synthetic, Kant relies on whether or not his mind could conceive of the predicate when given the subject. Ayer names this the "psychological criteria" (Ayer 78). To determine that a proposition is analytic, however, Kant relies on the principle of contradiction. Insofar as a proposition is true in that it cannot be A and ~A at the same time, it is necessarily analytic. Ayer criticizes Kant's definition because it creates a situation where "a proposition which is synthetic according to the former may very well be analytic according to the latter" (Ayer 78). While it is true that one could think of "7 + 5" without also be thinking of "12", it is also true that "7 + 5" necessarily add up to "12". This proposition "cannot be denied without self-contradiction" (Ayer 78), which allows it to exhibit the properties of both synthetic and analytic propositions as defined by Kant. As a revision to Kant's flawed definitions, Ayer therefore proposes that "a proposition is analytic when its validity depends solely on the definition of the symbols it contains, and synthetic when its validity is determined by the facts of experience" (Ayer 78).
In addition, Ayer points out that mathematical propositions are necessarily tautological, which would render these propositions analytic. A proposition that states "every eye-doctor is an optometrist" is tautological and analytic because as we understand the meaning of the two symbols "eye-doctor" and "optometrist", we find that they are synonymous. Similarly, after having gained knowledge of the symbol "7 + 5", we find that it is synonymous to the symbol "12". It is necessary true that "7 + 5" is "12", and that "12" is the definition of the symbol "7 + 5" (Ayer 85). In this case, the proposition "7 + 5 = 12" must be a tautology, as the predicate "12" adds nothing to the definition of "7 + 5". In response to Kant's idea that we must conform to our intuition to synthesize the numbers given in the subject, Ayer believes the fact that "most of us need the help
to make us aware of [the] consequences does not show that the [mathematical] relation
is not purely logical relation. It shows merely that our intellects are unequal to the task of carrying out very abstract processes of reasoning without the assistance of intuition" (Ayer 83). In an example with a set of larger numbers, the tautological properties of the proposition "91 x 79 = 7189" does not appear to us immediately. "To assure ourselves that "7189" is synonymous with "91 x 79", as Ayer indicates, "we have to resort to calculation, which is simply a process of tautological transformation" (Ayer 86). Kant is mistaken to think that calculations are synthetic processes, because they are in fact "process[es] by which we change the form of expressions without altering their significance" (Ayer 86). That is to say, calculations are simply alterations without adding anything to the original proposition. Hence, it is undeniable that mathematical propositions are analytic tautologies, even if we have to resort to relying on our intuition to perform calculations.
Let us conclude, then, that propositions of pure mathematics are analytic. With this criterion established with certainty, we can then combine mathematics with geometrical axioms to generate the system of geometry. Because both constituents are analytic, we can legitimately refute Kant's notion that geometry is synthetic. Thus, I have justified that Kant's hypothesis of the synthetic system of geometry is defeated by Ayer. As a wholly analytic system, geometry is therefore true in itself but cannot reveal for us anything in regards to the physical world. Therefore, it is also verified that Kant had made an invalid leap to derive the notion of physical space from geometry.
Ayer, A. J. Language Truth and Logic. New York: Dover Books Inc.
Kant, Immanuel. Critique of Pure Reason, Abridged. Trans. Werner S. Pluhar. Abrid. Eric Watkins. Indianapolis: Hackett Publishing Company, 1999.