Kant and Mathematics a Short Essay
Synthesis of Synthesis and a priori Concepts
This is a story of a 282-year-old man and his struggle to understand a science thousands of years older: Mathematics. If I were asked to name a philosopher who culminated and influenced the preceding centuries of the modern philosophy of mathematics, and yet be least understood, it would be Immanuel Kant. The 282-year-old man.
A philosopher's starting point is always knowledge—the central branch of philosophy being epistemology. (Logically speaking, metaphysics precedes epistemology; but to reach an understanding of metaphysics, one must first realize what establishes an “understanding” is and only a study of epistemology can help us.) Kant's starting point was his famous “analytic-synthetic distinction” in the introduction to the Critique of Pure Reason. Kant's motivation was largely a response to Hume-esque skepticism preceding his time—and called, by the end of the introduction, for a synthesis of the a priori - a posteriori dichotomy in epistemology in the form of a “synthetic a priori” knowledge.
To clarify terms, “a priori” refers to knowledge true before or independent of experience; “a posteriori” means knowledge predicated on experience. In section four of the introduction to Kant's Critique, “analytic” and “synthetic” judgments are defined: “Analytic judgments (affirmative) are therefore those in which the connection of the predicate with the subject is thought through identity; those i which this connection is thought without identity should be entitled synthetic” (48). “Analytic” therefore means statements true only when the predicate of a statement is contained in its subject; “synthetical” then means a statement where the predicate is not within the subject, depending upon a greater context to be true. “Greater context” means, necessarily, a context empirical in nature—synthetical judgments are strictly empirical in nature. “Judgments of experience, as such, are one and all synthetic.” (p. 49).
The economic way of describing synthetical and analytical judgments is that analytics are known a priori; synthetics known a posteriori. In essence, the differences between analytic and a priori are negligible and are essentially the same—the same is true for a posteriori and synthetic. (I attribute insistence from philosophers that the differences among these terms are substantial to philosophical tradition and not to any rational basis.)
“All mathematical judgments, without exception, are synthetical.” (Kant 52) Kant goes on after this statement opening section five of the introduction to the critique to show how all mathematical judgments are synthetical. Euclidean geometry, arithmetic, and presumably calculus were deemed synthetic—and by extension, empirical. Kant uses as an example the judgment that “7+5=12” to explain that no where within the concepts of “7,” “5” or even “+” (analyzed individually and combined) do not contain the concept of “12.” Thus to prove this statement, we must appeal to knowledge perceptual in nature; with graphing, for example. Later, Kant claims that only pure mathematics is synthetic a priori (55).
But before I proceed any further with Kant, I want to focus on the metaphysical aspect of the analytic-synthetic dichotomy. Its metaphysical presupposition is that the universe is composed of facts either necessary or contingent—that is, facts logically necessary or facts not logically necessary. If we were living in a necessary world, analytic knowledge would grant us knowledge of reality. If we were to exist in a contingent world, synthetic knowledge would be our window to reality.
Kant spends much time on the contemplation of a necessary existence—if reality is necessary, we have no means of knowing so, as only the contingent is perceptible. Only a priori knowledge can apprehend a necessary being, and because a priori knowledge cannot connect in any way to reality (as a posteriori knowledge does), we can only conclude a possibility of a necessary being. But the evidence exists for the contingency of reality in all other perceptions. The nominalism of Kant renders metaphysical contemplation impossible—interestingly, Kant claims in section five of the introduction that metaphysics is a priori synthetic.
What is important is Kant's effect on western thought from that point in history. Having resigned himself from the search for a synthetic-a priori understanding of existence by the end of his life (and having destroyed perhaps the only possible way in which such knowledge may be discovered, that is, through metaphysical theorizing)—he provided the clearest distinction between the necessary and the contingent. It is the metaphysical basis for the a priori-a posteriori, the metaphysical basis for the divorce of mathematics from the rest of the natural sciences with mathematics being an analytic science of the a priori (the idea that math is analytic is from the perspective of western culture, no Kant himself.) And thus we approach the “reflection” stage of this essay.
Although it is downright illegal for an undergraduate student to challenge Kant or to think he or she knows anything substantial or both (as in my case), I prefer to live dangerously and I will risk being honest.
I sensed that many of my fellow mathematics students were disturbed by the fictionalists—and probably even threatened to a certain degree by these thinkers. But in the grand flow of philosophy, Kant's philosophy has a far greater influence and a far more radical implication on the epistemology and ontology of mathematics than the fictionalists. For example, if we were to ask a person what they noticed to the greatest epistemic difference between the nature of mathematics and physics, they might say that physics deals with things (that it is “synthetic”) and mathematics deals with relations of ideas (Hume's actual term for a priori concepts.) This way of thinking is no doubt a product of all the philosophers after Descartes. But it was Kant drew the sharpest disctiontion between them and explicitly defined their metaphysical basis—and only one man is responsible when the next question we ask the person the ontological status of mathematics versus physics: The person won't know.
After all, there is no way of establishing a necessary being and thus no way to connect our non-empirical concepts to reality. The problem of induction becomes a real concern when we know we can only establish a contingent existence (which is phenomenal and not certain either.) We cannot apply a priori knowledge to a contingent world, and we cannot universalize a posteriori-synthetic concepts with absolute certainty as we can with pure concepts. In that regard, can we even establish an ontological status for physics? Our hypothetical person ought to be uncomfortable by this point.
The analytic-synthetic distinction lost its trendy status when Willard Quine wrote his “Two Dogmas of Empiricism” essay (which utilized linguistic philosophy, whose strongest basis is phenomenology—a philosophy inspired by and fully compatible with Kant's philosophy)—but the analytic-synthetic dichotomy lives on, seemingly unharmed by Quine. And other ideas considered antiquated regarding his philosophy of math is that he considered Euclidean geometry to be a priori synthetic and that it were The Geometry—but rather, Kant’s acknowledgement that geometry is chiefly a synthetic science, relying on a priori axioms, implies that Kant did not rule out (and possibly predicted) the later discovery of non-Euclidean geometry. Kant is still very much alive in this sense.
Kant haunts the world of philosophy and provokes the questions integral to the philosophy of mathematics: What is mathematics (ontologically speaking)? What epistemic status does math have, being an a priori science? And most importantly, is it even real? Many schools in the philosophy of mathematics are simply responses to these Kantian provocations.
And now for my so-called “sweeping generalizations” and flagrant “over-simplifications.” Is this Kant-centrism The Rational Alternative in our philosophical inquiries? Or must we re-examine how we got to our Kant-centrism? When has the a priori-a posteriori concepts been primary to epistemology?
The idea of a split between pure and empirical concepts can be traced to before the Enlightenment—but only after Descartes’ time can we see an explicit separation between the concepts. And each philosopher acknowledging the a priori-a posteriori distinction has treated the distinction as a philosophical primary not to be question—Kant himself presupposes it in the critique and in all other of his works.
It can be said that genius is often in the ability to prove or state the obvious, and with regard to the analytic-synthetic dichotomy, Kant comes short of genius (or perhaps deceivingly beyond genius.) The distinction has never received a serious critique or proof of necessity in epistemology. Also, the injection of “logic” into metaphysics with the necessary-contingent distinction has yet to be justified—there is no cause for us to evaluate reality in terms of logic; it is merely a twist on our quest to discover if reality is ours to understand.
What do I know of reality? First, it must exist apart from the mind—or else consciousness itself becomes impossible. As such reality’s independence from consciousness necessitates an absolute existence. Our connection to this reality is through our senses—which, once processed by the mind, provide perceptions of both quality and quantity. Perceptions are not in the object, but have no other source but reality itself to draw on. Any seeming alterations are due to the absoluteness of reality of the sense organs themselves, and are testaments to the accuracy of perception. Our initial concepts are perceptions without specificity—and beyond this are logically-coherent, but contextual inferences granted on virtue of our concepts and perceptions previously established to be correspondent truths.
Abstractions are a matter of cognitive science and epistemology. Concepts of quantity are derived just as those concepts of quality philosophers have spent much time analyzing (such as our concept of a chair.) What I see as the greatest difference between a science of the concepts of quantity (mathematics) and a science of concepts of quality (physics, biology, geology, et cetera) is with regard to theory-formation and induction, not metaphysical status.
Mathematics requires quantitative abstractions made early in life and relies on fewer observations than other sciences—but this by no means restricts math as a priori, if such a term were valid. Further abstractions must be made by logical inference, but such inferences do not separate mathematics from reality. Beyond this, little more can be said as scientific induction itself has been neglected due to acceptance of the necessary-contingent distinction, the analytic-synthetic distinction, and the problem of induction predicated on the previously stated distinctions.
As philosophers and mathematicians, the time has come to move beyond Kant and his time. To work with facts and not propositions; cognition and not the elusive, antiquated synthetic and analytic dichotomy; metaphysics and not the denial of it.
Kant, Immanuel. The Critique of Pure Reason. Trans. Norman Kemp Smith. New York: St. Martin’s. 1965.