The primary concern of mathematics is number, that is the positive integers.(Erret Bishop)
Mathematics is that portion of our intellectual activity which transcends our biology and our environment. The principles of biology as we know them may apply to life forms on other worlds, yet there is no necessity for this to be so. The principles of physics should be more universal, yet it is easy to imagine another universe governed by different physical laws. Mathematics, a creation of mind, is less arbitrary than biology or physics, creations of nature; the creatures we imagine inhabiting another world in another universe, with another biology and another physics, will develop a mathematics which in essence is the same as ours. In believing this we may be falling into a trap: Mathematics being a creation of our mind, it is, of course, difficult to imagine how mathematics could be otherwise without actually making it so, but perhaps we should not presume to predict the course of the mathematical activities of all possible types of intelligence.. On the other hand, the pragmatic content of our belief in the transcendence of mathematics has nothing to do with alien forms of life. Rather it serves to give a direction to mathematical investigation, resulting from the insistence that mathematics be born of an inner necessity.
The primary concern of mathematics is number, and this means the positive integers. We feel about number the way Kant felt about space. The positive integers and their arithmetic are presupposed by the very nature of our intelligence and, we are tempted to believe, by the very nature of intelligence in general. The development of the theory of the positive integers from the primitive concept of the unit, the concept of adjoining a unit, and the process of mathematical induction carries complete conviction. In the words of Kronecker, the positive integers were created by God. Kronecker would have expressed it even better if he had said that the positive integers were created by God for the benefit of man (and other finite beings). Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.
Almost equal in importance to number are the constructions by which we ascend from number to the higher levels of mathematical existence. These constructions involve the discovery of relationships among mathematical entities already constructed, in the process of which new mathematical entities are created. The relations which form the point of departure are the order and arithmetical relations of the positive integers. From these we construct various rules for pairing integers with one another, for separating out certain integers from the rest, and for associating one integer to another. Rules of this sort give rise to the notions of sets and functions.
A set is not an entity which has an ideal existence. A set exists only when it has been defined. To define a set we prescribe, at least implicitly, what we (the constructing intelligence) must do in order to construct an element of the set, and what we must do to show that two elements of the set are equal. A similar remark applies to the definition of a function: in order to define a function from a set \(A\) to a set \(B\), we prescribe a finite routine which leads from an element of \(A\) to an element of \(B\), and show that equal elements of \(A\) give rise to equal elements of \(B\).
Building on the positive integers, weaving a web of ever more sets and more functions, we get the basic structures of mathematics: the rational number system, the real number system, the euclidean spaces, the complex number system, the algebraic number fields, Hubert space, the classical groups, and so forth. Within the framework of these structures most mathematics is done. Everything attaches itself to number, and every mathematical statement ultimately expresses the fact that if we perform certain computations within the set of positive integers, we shall get certain results.
Mathematics takes another leap, from the entity which is constructed in fact to the entity whose construction is hypothetical. To some extent hypothetical entities are present from the start: whenever we assert that every positive integer has a certain property, in essence we are considering a positive integer whose construction is hypothetical. But now we become bolder and consider a hypothetical set, endowed with hypothetical operations subject to certain axioms. In this way we introduce such structures as topological spaces, groups, and manifolds. The motivation for doing this comes from the study of concretely constructed examples, and the justification comes from the possibility of applying the theory of the hypothetical structure to the study of more than one specific example. Recently it has become fashionable to take another leap and study, as it were, a hypothetical hypothetical structure—a hypothetical structure qua hypothetical structure. Again the motivations and justifications attach themselves to particular examples, and the examples attach themselves to numbers in the ultimate analysis. Thus even the most abstract mathematical statement has a computational basis.
The transcendence of mathematics demands that it should not be confined to computations that I can perform, or you can perform, or \(100\) men working \(100\) years with 100 digital computers can perform. Any computation that can be performed by a finite intelligence—any computation that has a finite number of steps—is permissible. This does not mean that no value is to be placed on the efficiency of a computation. An applied mathematician will prize a computation for its efficiency above all else, whereas in formal mathematics much attention is paid to elegance and little to efficiency. Mathematics should and must concern itself with efficiency, perhaps to the detriment of elegance, but these matters will come to the fore only when realism has begun to prevail. Until then our first concern will be to put as much mathematics as possible on a realistic basis without close attention to questions of efficiency.
Unfortunately the promise held out to mathematics by the arithmetization of space was not fulfilled, largely due to the intervention, around the turn of the century, of the formalist program. The successful formalization of mathematics helped keep mathematics on a wrong course. The fact that space has been arithmetized loses much of its significance if space, number, and everything else are fitted into a matrix of idealism where even the positive integers have an ambiguous computational existence. Mathematics becomes the game of sets, which is a fine game as far as it goes, with rules that are admirably precise. The game becomes its own justification, and the fact that it represents a highly idealized version of mathematical existence is universally ignored.
Of course, idealistic tendencies have been present if not dominant in mathematics since the Greeks, but it took the full flowering of formalism to kill the insight into the nature of mathematics which its arithmetization could have given.
To see how some of the most basic results of classical analysis lack computational meaning, take the assertion that every bounded non- void set \(A\) of real numbers has a least upper bound. (The real number \(b\) is the least upper bound of \(A\) if \(a \leqslant b\) for all \(a\) in \(A\) and if there exist elements of \(A\) that are arbitrarily close to\(b\).) To avoid unnecessary complications, we actually consider the somewhat less general assertion that every bounded sequence \(\{ x_k\} \) of rational numbers has a least upper bound b (in the set of real numbers). If this assertion were constructively valid, we could compute \(b\), in the sense of computing a rational number approximating \(b\) to within any desired accuracy; in fact we could program a digital computer to compute the approximations for us. For instance, the computer could be programmed to a sequence \(\{(b_k,m_k)\}\) of ordered pairs, where each \(b_k\) is a rational number and each nik is a positive integer, such that (i) \(x_j \leqslant b_k + k^{-1}\) for all positive integers \(j\) and \(k\), and (ii) \(x_{m_k} \geqslant b_k - k^{-1}\) for all positive integers \(k\). Unless there exists a general method \(Μ\) that produces such a computer program corresponding to each bounded constructively given sequence \(\{x_k\} \) of rational numbers, we are not justified, by constructive standards, in asserting that each of the sequences \( \{x_k\} \) has a least upper bound. To see the scope such a method \(Μ\) would have, consider a constructively given sequence \(\{n_k\}\) of integers, each of which is either \(0\) or \(1\).
Using the method \(M\), we compute a rational number \(b_3\) and a positive integer \(N \equiv m_3\) such that (i) \(n_j \leqslant b_3 + \frac{1}{3} \) for all positive integers \(j\). If \(n_N = 0\), then (i) and (ii) imply that \( n_j \leqslant b_3 + \frac{1}{3} \leqslant n_N + \frac{2}{3} = \frac{2}{3} \) for all \(j\)[...]
© Foundations of Constructive Analysis, Erret Bishop, p. 1 -10