To understand maths

What is the basis for reason? And mathematics?

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Ed Dirac
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To understand maths

Post by Ed Dirac »

Hi,

I just finished my first University year (mathematics), and I am writing this since my main interests on mathematics are metamathematics , the philosophy of mathematics and its foundations. I think that to study and analyze maths from this perspective I have to know how to work in certain areas of maths to understand how they function globally. That is why I would like you to guide me about what areas I should study before to study the aforementioned. I am quite lost on this issue so to clarify it a bit would be enough on your part.

Thank you in advance for your attention. A greeting.
Impenitent
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Re: To understand maths

Post by Impenitent »

mathematics at base is just another language...

study the philosophy of semiotics...

check out C.S. Peirce

-Imp
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Arising_uk
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Re: To understand maths

Post by Arising_uk »

I agree with Imp about C.S.P

Not quite sure what you're asking as I presume there are 'meta-mathematical' mathematicians writing out there?

But from a Philosophy of Mathematics point of view try,

Russell and Whiteheads tome, Principia Mathematica - too expensive to buy look in your university library.

Russells - Introduction to Mathematical Philosophy.

Not so much 'meta' but ground floor and basement, gave Godel an insight.

There's also Frege and Carnap.

Hope this is of some help.
Ed Dirac
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Re: To understand maths

Post by Ed Dirac »

Thanks you both.

I was looking for the maths areas that I should study before studying philosophy of mathematics. I think I am going to study formal logic, set theory and mathematical logic. But I didn't know about semiotic it looks something to take in account so at the same time I will study Pierce and semiotics.

Then I will study those books and authors Arising_uk mentioned.

I don't know if it is any difference between metamathematics and philosophy of mathematics, maybe are the same.

Why do you say that maths are another lenguaje? This is explained by Pierce?
Impenitent
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Re: To understand maths

Post by Impenitent »

semiotics is the study of symbols and their meanings

after Pierce, see Saussure (for semiotics... )

Why is mathematics a language?

Here are some definitions of language:
a systematic means of communicating by the use of sounds or conventional symbols
a system of words used in a particular discipline
a system of abstract codes which represent antecedent events and concepts [1]
the code we all use to express ourselves and communicate to others Speech & Language Therapy Glossary of Terms]
a set (finite or infinite) of sentences, each finite in length and constructed out of a finite set of elements Noam Chomsky.

These definitions describe language in terms of the following components:
A vocabulary of symbols or words
A grammar consisting of rules of how these symbols may be used
A 'syntax' or propositional structure, which places the symbols in linear structures.
A 'Discourse' or 'narrative,' consisting of strings of syntactic propositions [2]
A community of people who use and understand these symbols
A range of meanings that can be communicated with these symbols

Each of these components is also found in the language of mathematics.

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

-Imp
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Arising_uk
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Re: To understand maths

Post by Arising_uk »

Ed Dirac wrote:...
I don't know if it is any difference between metamathematics and philosophy of mathematics, maybe are the same.
...
Unfortunately not, 'meta-mathematics' is, I presume, using mathematical methods to study mathematics, philosophy of mathematics is using logic to study mathematics. So maybe my suggestions won't help.
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Arising_uk
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Re: To understand maths

Post by Arising_uk »

Impenitent wrote:semiotics is the study of symbols and their meanings

after Pierce, see Saussure (for semiotics... )
And after that there's Umberto Eco.
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Necromancer
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Re: To understand maths

Post by Necromancer »

Welcome Ed Dirac

Here may be some:
http://plato.stanford.edu/entries/nomin ... thematics/ - Nominalism in the Philosophy of Mathematics

http://plato.stanford.edu/entries/philo ... thematics/ - Philosophy of Mathematics

Cheers!

Necro- (I speak death, boo - boo :wink: )
Impenitent
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Re: To understand maths

Post by Impenitent »

Arising_uk wrote:
Impenitent wrote:semiotics is the study of symbols and their meanings

after Pierce, see Saussure (for semiotics... )
And after that there's Umberto Eco.
aye, The name of the rose was brilliant

-Imp
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Arising_uk
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Re: To understand maths

Post by Arising_uk »

Impenitent wrote: aye, The name of the rose was brilliant

-Imp
It was, although I think A Theory of Semiotics was better. :)
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A_Seagull
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Re: To understand maths

Post by A_Seagull »

Ed Dirac wrote:Hi,

I just finished my first University year (mathematics), and I am writing this since my main interests on mathematics are metamathematics , the philosophy of mathematics and its foundations. I think that to study and analyze maths from this perspective I have to know how to work in certain areas of maths to understand how they function globally. That is why I would like you to guide me about what areas I should study before to study the aforementioned. I am quite lost on this issue so to clarify it a bit would be enough on your part.

Thank you in advance for your attention. A greeting.
In disagreement with my learned friends......

Mathematics is an abstract logical system for generating theorems. It is based upon axioms and processes of logical inference which act upon logical entities. The problem is that the axioms and processes are not explicitly stated, instead they are implicit. For example "2+2 = 4" contains a lot of implicit rules about symbols and their relationships.
wtf
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Re: To understand maths

Post by wtf »

A_Seagull wrote:The problem is that the axioms and processes are not explicitly stated
The axioms of math are explicitly stated. The standard set of axioms is Zermelo-Fraenkel set theory, or ZF. https://en.wikipedia.org/wiki/Zermelo%E ... set_theory

The underlying system of logic is first-order predicate logic. https://en.wikipedia.org/wiki/First-order_logic

I am not sure where you got your notion that the axioms of math are not specified. On the contrary, they're specified in extreme detail.
A_Seagull wrote:"2+2 = 4" contains a lot of implicit rules about symbols and their relationships.
There's a lot of implied meaning, but it's implied in the sense of being left out for sake of compact notation. If challenged, any mathematician (or competent undergrad math major) could give you the precise set-theoretic definition of the symbols '2', '+', '=', and '4' that makes the statement true. In other words the symbols could be expanded out into the primitive language of set theory and proven directly from the axioms.
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A_Seagull
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Re: To understand maths

Post by A_Seagull »

wtf wrote:
A_Seagull wrote:The problem is that the axioms and processes are not explicitly stated
The axioms of math are explicitly stated. The standard set of axioms is Zermelo-Fraenkel set theory, or ZF. https://en.wikipedia.org/wiki/Zermelo%E ... set_theory

The underlying system of logic is first-order predicate logic. https://en.wikipedia.org/wiki/First-order_logic

I am not sure where you got your notion that the axioms of math are not specified. On the contrary, they're specified in extreme detail.
A_Seagull wrote:"2+2 = 4" contains a lot of implicit rules about symbols and their relationships.
There's a lot of implied meaning, but it's implied in the sense of being left out for sake of compact notation. If challenged, any mathematician (or competent undergrad math major) could give you the precise set-theoretic definition of the symbols '2', '+', '=', and '4' that makes the statement true. In other words the symbols could be expanded out into the primitive language of set theory and proven directly from the axioms.
Ok I challenge you then! (I presume you are a "mathematician or competent undergrad math major".) Using the axioms in either of your links, show how "2+2=4" can be proven or generated as a theorem.

I think you are going to struggle to do so as in a cursory read of the links I could find no reference to "2" , "4" or "+".
wtf
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Re: To understand maths

Post by wtf »

A_Seagull wrote: Ok I challenge you then! (I presume you are a "mathematician or competent undergrad math major".) Using the axioms in either of your links, show how "2+2=4" can be proven or generated as a theorem.

I think you are going to struggle to do so as in a cursory read of the links I could find no reference to "2" , "4" or "+".
The proof sketch is that you use the von Neumann definition of the ordinals to construct a model of the Peano axioms within set theory. In the Peano axioms, 2 is defined as S1, the successor of 1; and 1 is defined as the successor of 0. So 2 = SS0. The '+' operation is defined recursively in terms of the successor function. 4 is defined as SSSS0. The details of course are tedious.

https://en.wikipedia.org/wiki/Ordinal_n ... f_ordinals

https://en.wikipedia.org/wiki/Ordinal_n ... f_ordinals and in particular https://en.wikipedia.org/wiki/Peano_axioms#Addition

Naturally a cursory read of the Wiki page on Zermelo-Fraenkel set theory is not a substitute for obtaining knowledge of the subject. But you'll find the relevant definitions of '2', '4', and '+' in the links I gave. The definition of '=' is given by the axiom of extensionality on the Zermelo-Fraenkel page. It also has its own Wiki page. https://en.wikipedia.org/wiki/Axiom_of_extensionality

I am still wondering why you think mathematicians don't have an axiomatic foundation for their subject, when in fact the development of set theory as the foundation of math was one of the major mathematical projects of the 20th century. I recognize your handle from "that other philosophy forum" and you've always struck me as level-headed. Why are you making a demonstrably false claim? You can Google Cantor, Russell, Frege, Zermelo, and Gödel to get a sense of the development of set theory as the foundation of math in the late 19th and early 20th centuries.

I'm sure you've heard the story that Russell and Whitehead took several hundred pages to prove that 1 + 1 = 2. But prove it they did. https://en.wikipedia.org/wiki/Principia_Mathematica
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A_Seagull
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Re: To understand maths

Post by A_Seagull »

wtf wrote:
A_Seagull wrote: Ok I challenge you then! (I presume you are a "mathematician or competent undergrad math major".) Using the axioms in either of your links, show how "2+2=4" can be proven or generated as a theorem.

I think you are going to struggle to do so as in a cursory read of the links I could find no reference to "2" , "4" or "+".
The proof sketch is that you use the von Neumann definition of the ordinals to construct a model of the Peano axioms within set theory. In the Peano axioms, 2 is defined as S1, the successor of 1; and 1 is defined as the successor of 0. So 2 = SS0. The '+' operation is defined recursively in terms of the successor function. 4 is defined as SSSS0. The details of course are tedious.

https://en.wikipedia.org/wiki/Ordinal_n ... f_ordinals

https://en.wikipedia.org/wiki/Ordinal_n ... f_ordinals and in particular https://en.wikipedia.org/wiki/Peano_axioms#Addition

Naturally a cursory read of the Wiki page on Zermelo-Fraenkel set theory is not a substitute for obtaining knowledge of the subject. But you'll find the relevant definitions of '2', '4', and '+' in the links I gave. The definition of '=' is given by the axiom of extensionality on the Zermelo-Fraenkel page. It also has its own Wiki page. https://en.wikipedia.org/wiki/Axiom_of_extensionality

I am still wondering why you think mathematicians don't have an axiomatic foundation for their subject, when in fact the development of set theory as the foundation of math was one of the major mathematical projects of the 20th century. I recognize your handle from "that other philosophy forum" and you've always struck me as level-headed. Why are you making a demonstrably false claim? You can Google Cantor, Russell, Frege, Zermelo, and Gödel to get a sense of the development of set theory as the foundation of math in the late 19th and early 20th centuries.

I'm sure you've heard the story that Russell and Whitehead took several hundred pages to prove that 1 + 1 = 2. But prove it they did. https://en.wikipedia.org/wiki/Principia_Mathematica
OK, perhaps there do exist formal and explicit axioms that can generate all the theorems of mathematics, but they are not ones that I use or want to use.

Why invoke set theory as foundational? - it just seems to introduce unnecessary complication.
Why Peano's axioms? - If you didn't know that they were supposed to generate the natural numbers you wouldn't realise it from the axioms themselves.
What was the point of Russell and Whitehead taking several volumes of writing to 'prove' that "1+1=2"? To prove anything in a deductive system you need a foundation, an original axiom if you like, so why didn't they just start with "1+1=2" as their original assumption? Then the job would have been done in one line!

It seems to me that all these attempts to create a formal axiomatic system for mathemnatics are trying to prove something from nothing. But of course you can't do that. So instead, the assumptions are hidden within their axiomatic assertions.

Even if your axioms do provide a complete axiomatic system for mathematics, can you be sure that there is not a simpler set of axioms that do the same job but more efficiently? I doubt it.

So that is why I prefer to consider the axiomatic foundations for mathematics to be implicit rather than explicit. All an axiomatic system for mathematics needs to do is to define the relationship between the symbols of mathematics and how the true theorems of mathematics can be derived from these relationships.
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