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Do we want to say that if they changed, they would not be logic/mathematics (as we know) anymore? How can we be sure that there's no other type of logic/mathematics?

  • Short answer: we don't. Cf: Intuitionism, Dialethianism, etc. – jobermark Jan 18 '18 at 1:28
  • We do not say that and they change constantly, most notably at the turn of the 20-th century. This phrasing might refer to something like "there is only one reality and truth", but this usually applies equally to physics, biology, etc. See What are the differences between philosophies presupposing one Logic versus many logics? – Conifold Jan 18 '18 at 23:54
  • all the mathematics that we use are deterministic mathematics. In a question of more than one variable, we solve for one and hold the other variables constant. Our mathematics cannot solve for more than one variable at a time (non-deterministic). Examples of this are the classical physical problem known as the three body problem and chaos theory. All of our mathematics as such are approximations to explain the sensual universe. – Swami Vishwananda Jan 19 '18 at 10:35
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I like the responses addressing Aristotle's logic, so i'll provide a different perspective.

In any rigorous logical system, generally you have symbols, and rules for manipulating these symbols. The clearest analogy would be if we compared sentences to positions on a chess board. There are some arrangements of pieces which just don't make sense, because they are invalid. This is because there are rules on how you move pieces. Different pieces behave differently, like logical symbols and English words do. So in this sense, a true statement would be a legal arrangement on the board that can be arrived at by legal movements. However, we should ask, how do our starting pieces and positions influence positions we can reach. For instance, pawns cannot move backwards, so White can never have a pawn on it's first rank, as White pawns start on the second rank.

The initial positions of the pieces, and the type of pieces themselves, are analogous to our axioms. The axioms are initial statements believed/assumed to be true. In this sense, they are the first legal board positions, from which all others derrive.

Now, one can easily imagine chess on a 10x10 board, a 4x4 board, a board that has 8 ranks but infinitely many collumns, toroidal boards, higher dimensional spaces, chess with different types of pieces, or other games like Shogi and Go which are quite different alltogether.

We can imagine different logical systems, with different rules of inferences (https://math.stackexchange.com/questions/1201492/is-the-modus-ponens-is-an-axiom-in-formal-logic) and different primitive symbols. Are these "true" or just as "valid" as the conventional, classical logic we adhere to? This is a tough question. Many of these logical systems share a "truth-like" property we care about, "consistency". I can entertain the notion "If I were a penguin, I wouldn't be able to fly". Despite the initial condition, me being a penguin, isn't exactly "true", this statement still "makes sense".

Kurt Goedel, a logician and mathematician, composed his Incompleteness Theorems which address some fundemental properties of logical systems. I'd suggest reading up on those on other stackexchanges, but in short it states "any sufficiently complex logical system cannot be complete and consistent". Complete means that it can determine whether any statement (which plays by the rules and uses the right symbols) is true or false, and consistent just means "it makes sense" and nothing can have multiple truth values. If we apply this to our chess analogy, we can observe that there are many possible board games, so we may ask "why is ours special", and the answer is really "it isn't, but at least it works". Our logical systems might not be the only possible logical systems, but I believe it's pretty useful to have faith in our initial assumptions on how to make deductions.

This question often comes up in a mathematical analog in the form "Are our numbers the right way to do numbers". At the moment, there's a few really useful foundations of modern mathematics. Some mathematicians prefer Shogi to Chess, and a few mathematicians argue whether a Bishop should be worth 3 points or 3.25, whether knights behave in the right way, etc, but mostly they care about the midgame and the endgame. It "just so happens" with numbers like "3" and properties like "addition" that they have strong correspondences with reality in being able to count things, add velocities, etc. So due to the way reality is, it gives a lot more legitimacy to our current number system, and I would argue logical system too.

If maths and philosophy are the games we play, I think mostly mathematicians and philosophers care about how the game pans out, what strategies are useful, and less about specific rules. Though give this a read for a starter. https://plato.stanford.edu/entries/axiom-choice/

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First of all, I'd state that there isn't a consensus on "one logic/mathematics", throughout history different versions of logic and mathematics were presented, but I will not go deeper on that as it wasn't really the question.

Now for the question - most commonly when we talk about logic we are talking about Aristotle's categories for the logic basis - substance, quality, quantity, relation, place, date, posture, state, action, passion. Or another common logic categories - Kant's - 4 categories that split into 3 sub categories each: quantity- unity, plurality, totality. quality- reality, negation, limitation. relation- inherence and subsistence, causality and dependence, community. modality- possibility, existence, necessity.

Both Aristotle and Kant came up with the Categories to conclude the limits within our knowledge can be. Without these categories, they wouldn't have been able to further their philosophy and build logical arguments - meaning, in another words, they practically couldn't "talk".

Now, could have Aristotle or Kant create a different set of categories, hence practically changing their entire philosophy base? Technically, one could argue that the answer must be yes. But another can argue that these categories (as Kant's are practically a rearrangement and expansion of Aristotle's) are ultimately the "one true" set of categories, and we, epistemology-wise, can't have it any other way. Both statements might be true, and that might be ultimately dependent on your definition of logic - again, won't go deeper here because it isn't really the question.

So, for conclusion, when one says "one logic" one ultimately states that we epistemology-wise have one set of rules that we think by, and we can't have it any other way. Is it really the truth? Do we actually have only one set of rules by which we think? Do we even have a set of rules? Is it static? Is it dynamic? Those are all valid questions that were asked and answered (and asked again, et cetra et cetra) throughout the entirety of philosophical thought.

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Mathematics is a vast set of diverse practices, linked by similarities that are easy to 'see' but hard to define precisely, like trying to say exactly what is and isn't a game. It's just wrong to say there is only one kind.

Logic developed in three different places. Indian logic, which reached some of it's most sophisticated use by Nagarjuna and in shaping Mahayana Buddhism. Chinese logic, which was suppressed in it's early days as a challenge to state power and records of it barely survive, but seems to have been about the process of correct inference. And the more familiar Greek culture of logic.

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