Why don't we have consensus in more complicated areas of logic?

Question

When I once realised I don't really understand how and why proof by contradiction works, I started reading about it. And apparently I wasn't the only one who felt there's something wrong about it - constructivists and intuitionists do not accept the law of excluded middle. This is what shocked me the most - I thought logic is something that has been already settled a long time ago. It appears the truth is very different though. Logic is an attempt to describe and systematize the way we reason. And if some people accept the law of the excluded middle, and some don't, then it feels the same as if some people believed 1+2=3, and others 1+2=4. If we can all agree on how basic arithmetic works, why don't we have consensus in more complicated areas of logic?

Human thought is not based on logic, but logic is based on human thought. Do you believe we will eventually have a consensus in this area? People try to solve much more complex problems, such as P vs NP. But shouldn't we focus on the basic laws of reasoning first, especially if we have doubts in that field?

Mathematics exists because of the assumption we think in the same way. So it has to be the case, maths can't be based on our personal opinions or feelings. The existence of constructivism makes it look that way though. I'm writing these words assuming you will understand them the same way I do. Otherwise, What would be the point in writing this?

Answer 1

Several things might be important to note. The first is that formal logic is relatively recent, by mathematical standards: it's modern version is about a century old. On the other hand informal reasoning has been around for quite some time, and there is little doubt that we can do good mathematics with it, as others have noted.

The second is that while some may reject the use of the excluded middle or even the use of the principle of explosion (also known as ex falso quodlibet), the relationship between these principles of reasoning are well explored. In particular:

1. Classical logic is consistent if and only if intuitionistic logic is as well, as shown by the Gödel-Gentzen translation
2. The principle of explosion is consistent if and only if minimal logic (which removes this principle) is as well. See this paper for a nice survey.

The combination of 1 and 2 shows that there is no fundamental disagreement between these various formulations of logic, just about what the statements in various logics express. But this is more a philosophical question, and mathematicians are not necessarily interested in these.

Note that there are deeper disagreements about which principles are acceptable in mathematics, more or less related to the question of finitism. In this case, there is no hope for a clear resolution as above, since consistency of non-finitary systems can not be shown to be equivalent to that of finitary systems.

This last point is essentially the failure of Hilbert's Program for building a consensus about the consistency and completeness of mathematical foundations. However, despite the prima facie failure of this program, reverse mathematics and proof theory can be seen as a rigorous way to explore the different viewpoints and find a compromise.

My conclusion is that, while these are very interesting philosophical questions that are still being explored, there is no fundamental disagreement about what constitutes a sound mathematical argument.

Answer 2

I want to emphasize only a point, in order to give you some hints for reflections.

I believe that, history of science and of human knowledge shows us that is very very difficult to assert that there are "principles" totally immutable and indubitable.

See Euclid's Elements : Common Notions n°5 :

The whole is greater than the part.

For two millenia there were quite a great "consensus" about this kind of "assumptions".

But see Georg Cantor :

Cantor defined countable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable.

In rough words, the natural numbers are are proper subset of the rational, but we have that we as much rational numbers as natural ones.

A similar fact has been already discovered by Galileo Galilei with the so-called Galileo's paradox :

So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all the numbers, nor the latter greater than the former; and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities. [from Galileo Galilei, Dialogues concerning two new sciences (1638), transl. Crew and de Salvio, Dover. pp. 31–33.]

Answer 3

Classical two valued logic works well in mathematics and computer science, where a declaration that all statements are in principle either true or false is acceptable. The cost of this declaration is that only a limited class of grammatical, meaningful statements can be tested and evaluated using the methods of formal logic. This is a very large and important class, but not a comprehensive one.

Outside mathematics where the truth of statements is more a matter of observation and where the careful definitions needed by logicians and philosophers in order to ensure that statements are either true or false, the law of the excluded middle is subject to more dispute.

There are statements for which we lack information about whether they are true or false, there are paradoxical statements such as the liar paradox, there are statements that appear to be partly true, there are concerns about "necessarily" versus "contingently" true, and there are concerns about true versus provable.

There have been many logicians who have doubted the law of the excluded middle, and there are many approaches to extend logic to deal with more than the simple cases of "True" and "False" for statements.

There are multi-valued logics, the closely related concepts of fuzzy logic, intuitionistic logic, modal logic, relevance logic, and paraconsistent logic, all of which share some but not all of the assumptions and techniques of classical two-valued logic in attempts to deal with its limitations.

These various approaches overlap to a significant degree, but so far no one of them has established a clear superiority over the others, and they all have serious limitations compared with classical two-valued logic. There appears to be a strong professional rivalry among logicians of these various schools combined with a lack of understanding of competing methods and approaches. There seem to be few who believe that a more unified approach is either possible or interesting.

Answer 4

After you have worked with proof by contradiction for a while, you will wonder what the controversy was all about. It will seem like an indispensable tool and its use will become second nature. A few people have just decided not to use this tool for some reason -- like some people decide for reasons of their own not to use power tools in carpentry.

Answer 5

Logic by creation is not something magical -- it is a cleverly made (long time ago, that is why cleverly) SCANNING tool. This mental method allows us to SCAN huge variety of rather simple everyday(everyscience) life examples and problems. Because it is a SCANNING tool it needs information which is carefully prepared for it. It can not operate on chaos, same way as you can not run browser if you did not boot up your OS.

PROBLEMS start if you don't see what logic is and begin to misuse it. This what happens with your original question of interest. Prove by contradiction was designed for VERY well prepared sets of information where the total combinations can be clearly seen and exclude each other.

Logic is not the method to think. It is method to analyze data. If you have no data or poor data there is nothing to use logic for. Thinking happens when you prepare your mental/emotional data.