Why mathematical reasoning is built on two-valued logic?

Question

I have a very basic question.

How would you answer to somebody that is asking you why in mathematics we use two-valued logic as the very ground of math reasoning instead of some multi-valued logic?

Is the reason purely practical, but – at the same time – based on conceptual/philosophical reasons that go way back in time?

Any feedback will be much appreciated!

EDIT: I feel the best way to think about this question is as what you would answer to a skeptical student that is always ready to drop the study of mathematics altogether, but that you would really like to convince on the beauty of it.

Answer 1

Two-valued logic reflects our conceptualization of the world: we typically think of the world as a place where things are or are not the case. And that conceptualization often works, i.e. it allows us to make inferences and predictions that often come out true. So, it seems we are capturing something of significance about the world and how it works, and we can use it to great effect for many practical purposes. (indeed, the very fact that we conceptualize the world as such means that it we're getting something right, otherwise our brains would have rejected it a long time ago).

In this, logic is not any different from other branches of math or science: we come up with mathematical idealizations, and we see if they are applicable and useful to think and make predictions about the world about us. And even if they don't always work perfectly, as long as they work some of the time, and we get a pretty good sense as to when and where our idealizations apply or don't apply, then we'll use it.

Two-valued logic clearly works in this sense: it may not perfectly capture everything that's going on around us (think fuzziness, uncertainty, quantum weirdness, etc.), but it's pretty darn effective and useful in may real world situations. Its simplicity is of course another big plus.

Answer 2

Mathematics can be viewed as unfolding of the kernel concepts of number and space through generalisations and abstractions over various operations and relations involving only them. The knowledge of mathematics grows with increasing sophistication, but remains always faithful to the primitive grasp of the kernel which links it to the empirical world.

Since mathematics always returns on itself, each time, enriched further, the question whether mathematical innovations are discoveries or inventions is quite wrong-headed; an answer would be, at best, "both", briefly to say. By this nature, mathematics is actually one of the leading motivations for the philosophical doctrines of innate ideas and rationalism, which recur in one or other form unabatedly in the contemporary philosophy as well.

So, mathematics realises its own judgement of truth. Hence, it is free to expel anything standing against its rationality as falsity or fallacy, while mathematical truths remain incorrigible, for there can be no empirical fact to falsify them. In other words, it could be said that empirical facts can only verify them. Consequently, the dichotomy of truth and the rest (i.e., falsity) is inherent in mathematics.

To explicate this point, an analogy can be drawn to our eyesight. Our visual perception is limited, but we can translate/transform the sight of things that are not in our visual limits into our limits, anyway. We can even give a visual counterpart to abstract objects. However, all of those are eventually our ways of seeing things, whatever their origins might be. In case that we become aware of something which is invisible to us, but we think it must be, even though we think it is beyond our ability, we aim at devising a method to carry its sight over somehow into the domain of our ability. Thus, there is no seeing for us other than our seeing to correct us. In this respect, the frequent characterisation of primitive mathematical truths as "self-evident" is rather misleading; we do not indeed find them out that they are true, we think within them, and there is no without. This aspect has led some philosophers to propound the view that mathematics actually describe an ontology (see, for example, Alain Badiou).

A clear example of the incorrigibility and the accumulative progress of mathematics associated with it is the history of Euclidean geometry. It has been understood that different geometries could be constructed, its presentation has been modified to fit in the current standards of rigour and precision; nonetheless, it has manifested its own development and remained a constituent of mathematical knowledge on a par with others. In this connection, Einstein's 1921 address titled "Geometry and Experience" would be a mind-broadening read.

How does this line of thought fare with intuitionism and constructivism in mathematics? We cannot go into the discussion of these topics, but it should be remarked that they are not about mathematical truth/falsity per se, but basically about the methods to attain and judge mathematical truth. In a similar vein, conjectures make an example for the pragmatical side of mathematical activity, rather than its essential theory.