Is mathematics religious?
What a strange question, I hear you say!
But if everyone's beliefs about everything are shaped by their own religious beliefs, as has been claimed, then that must include mathematics.
You (both of you) may recall the basic shape of the argument. Religious beliefs are beliefs about what is independently and unconditionally real (Chapter 2). And our beliefs about what is fundamental about reality will affect the kinds of hypotheses we will entertain when we are thinking about any particular area of reality, including mathematics (Chapter 4).
Here are some examples.
1. The Number-World Theory (Pythagoras, Plato, Leibniz). In this theory,
the numerals and other markings of mathematics stand for real entities in another world or dimension of reality (p.133).
In order for this to be true,
it would have to be the case that the quantitativeness of things relates to the other kinds of properties and laws true of them by being utterly independent of them all. Thus, the quantitative aspect is (at least part of) what things and their other kinds of properties depend on for existence (p.134).
The quantitative aspect as one of the aspects of our experience mentioned in Chapter 4, and is one of fifteen aspects identified by Herman Dooyeweerd. (The full list given in Clouser's book is: fiduciary, ethical, justitial, aesthetic, economic, social, linguistic, historical, logical, sensory, biotic, physical, kinetic, spatial and quantitative.)
Examples of the practical difference these religious beliefs have had on mathematics have been the resistance of the Pythagoreans to the idea of irrational numbers, and that of Leibniz to the idea of negative numbers.
There are more practical differences when we consider the intuitionists (Brouwer, Weyl, Poincaré), who make even the logical aspect to be dependent on the quantitative aspect. This forces them to deny "the existence of actual infinite sets" and therefore to "reject an entire branch of mathematics, the theory of transfinite numbers developed by Georg Cantor" (p.141).
2. John Stuart Mill, whose "theory was that numerals symbolise sensory perceptions" (p.134).
Mill defended this view of math[s] by arguing that not only the quantitative aspect, but all other aspects of our pre-theoretical experience are actually identical with its sensory aspect. That is, Mill's theory was that the nature of all reality is sensory (p.134).
This approach is similar to the number-world theory in selecting one (or two) of the aspects, and asserting that it is the non-dependent reality on which all of the other aspects depend.
3. Bertrand Russell, who took the logical aspect to be non-dependently and unconditionally real:
The logical laws, he says, are not only those to which all reality — actual or possible — must conform, but they are "the heart and immutable essence" of all things (p.144).
Thus mathematics "is nothing other than a short-cut way of doing logic" (p.135).
4. Instrumentalism, e.g., John Dewey, for whom the physical-biological aspect(s) have the non-dependent and unconditional reality:
through all his theorizing, he regards all other aspects as dependent on the physical-biological and never regards them, in turn, as dependent on anything else (p.144).
Under his theory, "humans are to be understood as essentially biological beings struggling to survive in a certain environment", and thus "all human cultural products are instruments" (p.136), helping us to survive.
Just as it is inappropriate to ask whether a hammer is true or false, it is equally inappropriate to ask that of mathematical tools. 1 + 1 = 2 is thus neither true or false, says Dewey, though it performs certain tasks well (137).
5. Belief in God, "which should lead us to the view that no aspect of creation is self-existent, nor does any generate any other since all are dependent on God alone" (p.145). Under this view of mathematics,
the abstractions we arrive at, numbers, sets, etc., will never be seen as independently existing realities. The are never more — or less — than the properties, relations, functions, etc., of the quantitative aspect true of the things and events of ordinary experience (p.146).
In summary, it would have been good to explore the practical implications of these different views more fully. While all of the views differ greatly on their understanding of the nature of mathematics, it is so far clear only in the cases of the number-world theory and of intuitionism that any specific theories of mathematics are affected by one's religious beliefs. But, for example, if I switch between the theories of Mill and of Russell, would I then be forced to change my beliefs about any specific theories of mathematics?
But I think enough has been said to demonstrate the point: that religious beliefs do indeed exert a controlling influence on mathematical theories.
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