Here's the final draft of my paper on mathematical statements as analytic statements. I'm very pleased with how it turned out. I didn't spend as much time on how identity statements are analytic as I would have liked, but I think that I did focus on things that were within the scope of a 4-ish page paper. Let me know what you think.

Mathematical Statements as Analytic

I claim that statements of mathematics are analytic. This challenges Kant’s view of mathematical statements. First, I will try to show why Kant’s reasoning may be contradictory. Then, taking each of his criteria independently, I will examine them in relation to mathematical statements. Finally, I will identify and address likely flaws in my arguments.

Allow me to point out a potential contradiction in Kant’s argument. Kant defines analytic and synthetic statements as follows: an analytic statement is a statement in which the predicate adds no new content to the concept of the subject; a synthetic statement is a statement in which the predicate does add new content to the concept of the subject (Ayer, 77). In the example “7+5=12”, he reasons that, because the reader can conceptualize “7+5” without conceptualizing “12” and vice versa, it must be synthetic. Finally, in the example, “all bodies take up space”, he argues that the truth of “all bodies take up space” is subject to the Law of Non-Contradiction (Ayer, 77). It would seem that, according to Kant, all three of these criteria are equivalent. Here, then, is a contradiction. Assume that “7+5=12” is synthetic by Kant’s reasoning. However, it cannot be true that “7+5=12" and 7+5=(not)12.” So, the truth of the statement is subject to the Law of Non-Contradiction. Then, it must be analytic. However, the statement cannot be both analytic and synthetic. So, if Kant is asserting that his three criteria are equivalent, then he appears to contradict himself (Ayer, 78). For the sake of this paper, I will consider all three of his criteria, in the hopes of showing that, regardless of which is chief among them (which is outside the scope of this paper), mathematical statements will be held analytic.

Let us first examine the psychological criterion: the subject and the predicate can be conceptualized independently in a synthetic statement and not independently in an analytic statement (B/F, 140). What does it mean to conceptualize objects such as “7+5” and “12”? I maintain that the concept of the complex object “7+5” is the union of the concepts of the objects “7”, “+”, and “5”. What, then, is the concept of the object “7”? I claim that we have no way of conceptualizing “7” without conceptualizing fundamental numbers and operations. Conceptualize an integer “n”. Now, conceptualize the integer immediately following “n”. Of course, you probably thought of “n+1”; then, in order to conceptualize any number, we must rely on at least addition and “1”. What about, the reader may protest, very large numbers, say 792? We have no way to conceptualize the object “792” as a value independent of its component values; when we think of “792”, we really think of “”7(10^2)+9(10^1)+2(10^0)”. That is to say, we cannot conceptualize any number without at the same time conceptualizing addition, multiplication, exponentiation, and the numbers “0”, “1”, and “10” (and division, for rational numbers, etc.). Then, by this argument, when we examine the statement “7+5=12”, the ideas of “7”, “+”, and “5” are all entailed in “12”, because we must conceptualize repeated addition of “1” for “7”, “5”, and “12”. It would appear, then, that the statement is analytic.

Next, let us examine the logical criterion: the relation between the subject and predicate is subject to the Law of Non-Contradiction in an analytic statement and not in a synthetic statement (Ayer, 77). Kant gives the statement “all bodies are heavy” as an example of a synthetic statement that fails this criterion, and “all bodies are extended” as an example of an analytic statement that fulfills this criterion. Certainly, it is possible for a body to be heavy and not heavy, whereas it is impossible for a body to take up space and not take up space. However, as I have pointed out, it is impossible by the definitions of the constituent symbols that “7+5=12” and “7+5=(not)12” are both true statements. Then, the statement would again seem to be analytic.

Finally, we come to Kant’s original definitions: an analytic statement is a statement in which the predicate adds no new content to the concept of the subject; a synthetic statement is a statement in which the predicate does add new content to the concept of the subject. This is a criterion of neither psychology nor logic, but, rather, of semantics (B/F, 140). That is to say, what the reader knows is not relevant to this distinction; what are relevant are the definitions attributed to the subject and predicate. Is there content in the predicate that is not present in the subject? What exactly does Kant mean by “content”? I hold that “content” refers to the set of properties attributed to an object. Then, all “bodies” have the property “being extended” but not “being heavy.” How does this help us make sense of the contents of “7+5” and “12”? What sort of property could be attributed to “12” that is not attributed to “7+5” or vice versa? The canny reader might say that “7+5” has the property of being two numbers added together, whereas “12” is only one number. However, since “12=7+5”, “12” has the property of being two numbers added together, as well; similarly, “7+5” has the property of having one value, namely, “12”. I claim that, because “7+5” and “12” are related through an identity, no new content can possibly be added to the subject by the predicate. The set of properties of “7+5” is the same as the set of properties of “12”, so “12” can attribute to “7+5” no new properties. Observe that this is without regard to what the reader comprehends; the reader might be under the impression that “7+5=0”, but, still, by the definitions of the constituent symbols, no new content can possibly be added to “7+5” by “12”. And, again, the statement would appear to be analytic.

My reasoning is not without flaws. In the case of the psychological criterion, one might note that irrational numbers cannot possibly be conceptualized through fundamental numbers and operations, because they cannot be expressed as one integer over another non-zero integer. However, they can be expressed as an infinite sum of rational numbers, and, since (it seems to me) we cannot conceptualize “infinity,” we cannot conceptualize a number that is defined as an infinite sum of numbers. Then, we can only conceptualize approximations of irrational numbers. However, if this is true, is it also true that “pi=3.1415” since our concept of pi is 3.1415 (or more digits, depending on one’s accuracy)? Certainly not, regardless of to how many decimal places you extend the predicate. So, it seems that we have some concept of pi that exceeds the aid of other fundamental numbers and operations, puncturing a hole in my argument that all numbers must be conceptualized as such. In the case of the semantic criterion, it could be noted that, however much I want to eschew the reader, there must be one. A statement cannot be viewed purely in terms of its semantics; the reader always brings conceptualizations to it. We cannot escape psychology. I have little response to this attack. Let us assume that the statement “7+5=12” is presented to a person who does not understand the constituent symbols. By the argument for the semantic criterion, they need not be able to for the statement to be analytic, since the statement is true by definition of those symbols. However, this regresses: there must be someone who knows the definition of the symbols, someone who can conceptualize them, or else the statement would be without meaning. I admit that this is a dire flaw in my argument for the semantic criterion, and leads down epistemic paths beyond the scope of this paper. Finally, my claim stated, “Statements of mathematics are analytic,” yet I have only taken the example “7+5=12.” The mathematically-minded reader should observe that, in order to fully demonstrate the veracity of my claim, I would need to also examine statements of inequality and functions (especially functions that are not one-to-one or onto). However, this is again beyond the scope of this paper.

I believe that I have adequately showed how Kant’s reasoning leads to a contradiction if one assumes that his semantic, logical, and psychological criteria are equivalent. Then, taking them to be separate, I hope to have given the reader cause to believe that mathematical statements are analytic by each criterion. It is outside the range of this paper, however, to explore whether Kant is truly assuming the equivalence of his criteria, and, if he is not, to choose which shall hold priority in making the distinction between analytic and synthetic statements.

Works Cited

Ayer, Alfred Jules. Language, Truth, and Logic. New York: Dover Publications, Inc, 1946.

Baggini, Julian, and Peter S. Fosl. The Philosopher’s Toolkit. Malden: Blackwell Publishing, 2003.