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Where is the core principles that govern mathematics?


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For example I asked around the proof of for 1 + 1 = 2 or mathematical proof for any addition and people mentioned different mathematicians for the proof. What is like the standard that is used as the foundation of mathematics. I took high school math but that is about it, I know from a little reading that there are axioms which I think are the foundation of math. Is there a place where these axioms are collected into one?

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9 hours ago, Alex Mercer said:

For example I asked around the proof of for 1 + 1 = 2 or mathematical proof for any addition and people mentioned different mathematicians for the proof. What is like the standard that is used as the foundation of mathematics. I took high school math but that is about it, I know from a little reading that there are axioms which I think are the foundation of math. Is there a place where these axioms are collected into one?

Another member started a project to look for something similar.

There is much useful material in this thread for you.

 

 

I will will draft a further response and add a list of books as requested in due course.

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I think that guy is looking for like a math of math, I'm wondering if their is a commonly known book where the axioms and proof that are the foundation for all math is. Does it exist or is this sort of stuff spread through various books

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7 minutes ago, Alex Mercer said:

I think that guy is looking for like a math of math, I'm wondering if their is a commonly known book where the axioms and proof that are the foundation for all math is. Does it exist or is this sort of stuff spread through various books

Understanding exist's, on the boundary of knowledge... 

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Keep in mind that in one sense, most mathematicians don't ascribe an enormous amount of importance to mathematical foundations, but in another they very much do:

The standard place where most mathematicians place the foundations of mathematics is set theory, most commonly in the form of Zermelo-Fraenkel with the axiom of Choice (or ZFC). I say standard because there is quite a bit of work on nonstandard foundations - simply removing the axiom of choice is common; some replace ZFC with Homotopy Type Theory (which is beyond me entirely). I say most because there are some mathematicians that don't work in ZFC at all - some even reject the consistency of ZFC (some even reject the consistency of ZF, though that's rare).

The sense in which it's not important is that for the most part, mathematicians don't need to know exactly how every symbol works; as long as certain things can be done, most mathematicians don't need to care what underlies it. The sense in which it is very important is that it's still necessary to know (or at least, strongly believe) that it can be done consistently; a failure in the foundation would be a failure of mathematics as a whole. So imagine it like the foundation of a building: it's not important that you know the details of the foundation, but it's important that you know the foundation is there doing its job.

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On 3/11/2021 at 5:53 AM, Alex Mercer said:

For example I asked around the proof of for 1 + 1 = 2 or mathematical proof for any addition and people mentioned different mathematicians for the proof. What is like the standard that is used as the foundation of mathematics. I took high school math but that is about it, I know from a little reading that there are axioms which I think are the foundation of math. Is there a place where these axioms are collected into one?

Dear Alex Mercer ! 

Why do you think we teach and learn philosophy of mathematics  ??

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3 hours ago, Alex Mercer said:

I think that guy is looking for like a math of math, I'm wondering if their is a commonly known book where the axioms and proof that are the foundation for all math is. Does it exist or is this sort of stuff spread through various books

Hello again Alex, I thought you had abandoned this thread.

I didn't suggest Aline was looking for the same thing as yourself, just that there were some ideas and information there that would be relevent and useful to your question.

 

So back to your question.

I am sorry to tell you that not only does it not exist but it is impossible for it to exist.

 

Just a little bit of history, then some references to what can be obtained (and appreciated) by someone with your stated mathematical background.

As the 19th century turned into the 20th four of the top mathematicians in the world tried to do exaxactly what you are asking.

All four failed for different reasons.

The first two worked together to try to produce a book which took its inspiration from Newton's famous Principia for Physics and was to be the equivalent for Mathematics,
So they called it Principia Mathematica.
Their book was produced and was indeed a massive effort and success, it was not comprehensive.
But by this time the scope and depth of Mathematics had grown so much that it was well beyond two people to comprehend it all, let alone one.
And Mathematics was, and still is, growing at an ever accelerating pace.
The two were Russell and Whitehead.

Meanwhile Klein had introduced the Erlangen Program which married geometry and algebra in an axiomatic way updating Euclid.

https://en.wikipedia.org/wiki/Erlangen_program

And Hilbert attempted to build on this to provide an axiomatic basis for all of Mathematics.

https://en.wikipedia.org/wiki/Hilbert's_program

Then along came Godel in 1931  who published (over a period of time) his completeness and incompleteness theorems.

He had proved that questions could be posed for all system of axioms as complicated or more complicated than simple arithmetic, questions that could not be answered within the system.

https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

 

With reagrd to you desired book list I would recommend the following

What is Mathematics, and elementary approach to ideas and methods

By Courant and Robbins

Principles of Mathematics

By Allendoerfer and Oakley

Unknown Quantity

By John Derbyshire

Discovering Modern Algebra

By Gardner

A survey of Modern Algebra

By Birkhoff and MacLane

From Geometry to Topology

By Graham Flegg

Introduction to Topology and Modern Analysis

By Simmons

Beginning Logic

Lemmon

Elementary Geometry

By Roe

Note this selection is far from comprehnsive, huge areas of maths are omitted entirely eg statistics and numerical methods theory.

But they will take you from the classical high school notation to modern notation, without which knowledge you would be struggling.

They would also lay a foundation for further studies at higher level.

 

Edited by studiot
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I should also add that these books were selected as beings suitable for self study and for those who are not following a formal course.

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