Strict finitism

Infinity in mathematical theories has always been a major source of controversy. We can only perceive a finite thing in its entirety, and this is true for our imagination too. With respect to intuition, this means we can only guess what a real infinity would look like in terms of our finite perception, and whether there exists infinity of any kind (actual or potential) at all. To eliminate this guesswork, theories in strict finitism are free from infinity.

In classical mathematics, one can state that the formula n2-1=(n+1)·(n-1) is true for all natural numbers. In strict finitism, the corresponding statement is that any concrete equation that matches the above formula is true. In other words, we only state that if someone gets hold of a concrete natural number, say, 19, then the resulting concrete equation, 192-1=(19+1)·(19-1), will be true. That is, while in classical mathematics the statement refers to all the elements of a (hypothetically) existing infinite set, in strict finitism it's merely an abstraction of concrete individual occurrences.

Infinite sets of classical mathematics may have counterparts in strict finitism. For example, the "set" of even numbers is basically defined as the property of being even. If a concrete number has this property, we say that the number is an "element of the set". So it's not really a set in the classical sense, but rather a common property that ties concrete occurrences together.

A "sequence" of rational numbers can be defined as a (finite) method that expects a single input, a natural number, and is guaranteed to produce a rational number as output in finitely many steps. Such methods can then be used to represent real numbers by finitary means.

Strict finitism is usually coupled with constructivism (constructivism taken to the next level, so to speak), but nevertheless it's possible to develop non-constructive theories that do not make use of infinity.(9)

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(9) Does strict finitism deny the existence of infinity?
No, not necessarily. It rather says that we don't need to care whether infinity exists or not, because we can still do useful mathematics without it. As of today, the resulting theories seem sufficient for the purpose of modeling the finite world we know about.

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