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He invented a field of mathematics now called Galois fields,
We know there is molecules right next to the line and they do not tend to zero. If this wasn't so there would be no tiling of floors
But infinitely thin means invisible or worse: non-existing - unintuitive.
Yes a line in mathematics has no width, that means it is an abstraction.
If abstract thinking is a problem for you then mathematics in general will be a problem for you.
Actualy, mathematic is unituitiv.
Quote from: Deecart on 29/07/2022 14:12:03Actualy, mathematic is unituitiv.Why do you say that?I find it very intuitive
Here's an old mathematical joke.How do you get an elephant into a jam jar?Consider an elephant outside a jam jar, then perform a trivial inversion of vector space.
If the "reality" of mathematics was in doubt, we would still be living in mud huts.
Why do you say that ?
The question is : "Is mathematic unintuitiv ?" (pretty simple question), not if "you" or someone else think it is good or bad, or even if it is the right way or the wrong way to do mathematic.
Some aspects of mathematics are unintuitive. For a lot of human history, zero, negative numbers and imaginary numbers weren't even conceived of.
I should say that it’s not quite true that the Babylonians didn’t have a symbol for zero. In Plimpton 322, the tablet we studied in class, there are some gaps between numerals that represent zeros in the middle of a number, the way the 0 in 101 represents zero tens. Later, they added a symbol for zero, but it was only used for zeroes that were in the middle of the number, never on either end. That way they could tell the number 3601, which would have been written 1,0,1, from 61, which would be written 1,1. But 60 and 1 would always be written identically. They never made the leap to using a zero symbol at the end of a number to eliminate the ambiguity completely. The oldest documented zero is surprisingly modern: it’s in a temple in India, and it dates from about 875 CE.One of the strange consequences of the lack of zero comes up in reciprocals. We usually think of reciprocals as number pairs like 2 and 1/2 that multiply together to equal 1. But Babylonian reciprocal tables, which made calculations quicker, listed any two numbers that multiplied to a power of 60 as “reciprocals.” For example, 5 and 12 are “reciprocals” in this sense because they multiply to 60. Why would this definition of reciprocal make sense? Because when you’re writing in base 60 without a zero, 60 looks just like 1! So do 1/60, 3600, and any other power of 60. It would be like us thinking of 4 and 25 as reciprocals because they multiply to 100. It would make some sense only if we didn't have a zero, so 1 and 10 looked the same.
You mean 0 was not understand by babylonian scientists ?
Because i think intuition has something to do with reality.Instead, mathematic is "a chimera" (build from various things that do not belong together in reality).
I have just read that :Quote from: alancalverdHere's an old mathematical joke.How do you get an elephant into a jam jar?Consider an elephant outside a jam jar, then perform a trivial inversion of vector space.This should explain what i think about "the reality" of mathematic.
Quote from: paul cotter on 29/07/2022 15:58:47If the "reality" of mathematics was in doubt, we would still be living in mud huts.No.You are here talking of physic who rely on SOME usage of mathematic.
Intuitive has nothing to do with reality. Intuitive means not requiring proof and much of maths is intuitive.
There Are Many LogicsWhen we examine the semantics of natural language we see enormous diversity in the kinds of expressions that we can generate. These are often a stimulus for new developments in logic, as researchers try to invent new formal languages (or modify old ones) that do a better job of capturing the semantics of particular aspects of natural language.What you study in a first class in symbolic logic is actually a specific type of propositional logic — classical, truth-functional propositional logic.Other logical systems have been developed that are “propositional” in that they deal with logical relations of propositions take as wholes, but they may not be “classical” or “truth-functional”.In classical logic we assume that propositions can only be true or false, not both, and there are no third options available. Non-classical logics drop or modify one of these basic assumptions.There are systems called “intuitionistic logics”, for example, that were designed to model the logic of “constructive provability”. We don’t need to go into the details of what that means, but it turns on the distinction between knowing that a statement is “true” and knowing that one can actually construct a proof — a derivation — of that statement.In classical logic you can prove that certain statements are true, without actually having to construct a proof for them. We can do this, for example, by showing that the negation of a statement, not-P, leads to a contradiction, and from that fact jump to the conclusion that P must be true. In classical logic, any statement that entails a contradiction must be false, and by double negation we see that not-(not-P) = P, i.e. that P must be true.For various reasons the founders of intuitionistic logic wanted to disallow proofs like this.
In the example given, the line, no proof is required as it is a defined concept which is obvious - if the line has thickness then it is a rectangle or rod.
Similarly, a+b=c is intuitive, indeed to try and prove it can be very difficult.