Numbers: Their Tales, Types, and Treasures.

Chapter 11: Numbers and Philosophy



The great German philosopher Immanuel Kant (1724–1804) was less concerned about the reality of abstract objects than he was in statements and propositions about these objects. Consider the following statements concerning the role of mathematics in the physical world:

A:Mathematics is rooted in observations and knowledge about the physical world, and its results tell us something about empirical reality.

B:Mathematics is a system of propositions, each of which is true in itself and needs no empirical verification or confirmation.

Kant distinguished between a priori knowledge and a posteriori knowledge. Knowledge is a priori if it is independent of any experience about the physical world (such as “all triangles have three sides”). Knowledge is a posteriori if it depends on empirical evidence (such as “there are six items in that box”).

A statement is called analytic if it is true in itself—for example, “all husbands are married.” This statement can be seen to be true because the word husband only refers to a married person. Understanding the meaning of the words is sufficient to judge the truth of the sentence. A statement where the meanings of the words alone do not imply whether the statement is true or false is called synthetic. The statement “all husbands are happy” (whether or not it is true) is synthetic because the word husband alone does not imply happiness. It appears that analytic statements are not very interesting, because they can be made in advance (that is, “a priori”), without referring to anything that is not already contained in the definition of the words in that statement. Synthetic statements seem to be more interesting because they make a claim that is not self-evident and that does not already follow from the meaning of the words in the statement. The correctness of a synthetic statement cannot be inferred just by analyzing its content. From this we see that synthetic statements tend to be a posteriori. One typically has to refer to experience and observation for determining the truth of a synthetic statement. Kant's big question now is whether any synthetic knowledge exists a priori and whether a mathematical statement such as 5 + 3 = 8 would be synthetic and a priori.

As a mathematical result about numbers, 5 + 3 = 8 is a statement that follows logically from the structure of the number sequence (1, 2, 3, 4…). As such, it is true because of the way mathematicians draw conclusions from the axioms and because of the way they use the rules of logic to determine the truth of statements. Assuming that the sequence of symbols (1, 2, 3, 4…) has the properties required by the Peano axioms, the truth of arithmetic statements follows inevitably from elementary rules of logic. The statement thus represents a priori knowledge and it is analytic because it just expresses a formal property of the ordered sequence to which the symbols 3, 5, and 8 belong.

As a statement about cardinal numbers, “5 + 3 = 8” means that if we combine two sets with cardinalities 5 and 3 (that is, if we form the “union” of these sets), we will obtain a set with cardinality 8. The truth of this result can also be verified in reality, where we can check by counting that 5 objects combined with 3 objects indeed gives a set with 8 objects. As a statement about empirical reality, it seems to be a synthetic statement.

The statement “5 + 3 = 8” is therefore obtained a priori, just by logical derivation from the axioms defining the properties of the number sequence, and yet it appears to be a synthetic statement that tells something about the physical universe. In view of our considerations on the psychology of numbers, it may indeed be doubted whether anything about this statement is a priori. The concept of number is clearly rooted in elementary knowledge about properties of the universe—knowledge that is, in part, acquired through evolution and innate, and in part culturally acquired. The same could be said about the logical rules, which are by no means totally self-evident (as we notice in the discussion about the rule of the excluded middle between intuitionists and classical mathematicians). But the logical rules are probably also, in part, rooted in core-knowledge systems, ingrained in our brain by evolutionary processes, and thus reflect some elementary properties of the (causal) mechanisms in the world that surrounds us.

The philosophical position that mathematics is not a priori but that all its objects have their origin in empirical knowledge is called empiricism. According to this view, mathematics is, after all, not so different from other natural sciences. American philosopher Willard van Quine (1908–2000), an important proponent of empiricism, said that mathematical entities, like numbers, exist as the best explanation for experience. Thus, mathematical results, like 5 + 3 = 8, are not completely certain, because they refer to observations that, at least in principle, could be wrong. Fortunately, mathematics is very central to all of science, and a large web of trusted knowledge depends on it, and thus it would be extremely difficult to change mathematics. This gives the impression that the results of mathematics are completely certain and not likely to be revised.

Indeed, mathematical considerations of much higher complexity than mere additions are routinely and successfully applied to predict phenomena in the physical world. People have often wondered how it is possible that abstract mathematics is so successful in describing reality.

Albert Einstein (1879–1955), in an address given in 1921 at the Prussian Academy of Sciences in Berlin, formulated this problem as follows: “How can it be that mathematics, being, after all, a product of human thought, which is independent of experience, is so admirably appropriate to the objects of reality?”18

Later, in 1960, Hungarian physicist Eugene P. Wigner (1902–1995) coined the expression of “the unreasonable effectiveness of mathematics in the natural sciences.”19 It is obviously a question that has remained a topic of discussion among mathematicians, natural scientists, and philosophers. In the introduction to the 2007 edition of The Oxford Handbook of Philosophy of Mathematics and Logic, American philosopher Stewart Shapiro (1951–) says, “Mathematics seems necessary and a priori, and yet it has something to do with the physical world. How is this possible? How can we learn something important about the physical world by a priori reflection in our comfortable armchairs?”20

Einstein, in his 1921 address, attempts to give an answer: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”21 This expresses the point of view of an empiricist and applied mathematician who is reluctant to accept that mathematics has any a priori relevance for the physical world. To the applied mathematician, any application of mathematics to reality can be understood as a process of mathematical modeling.