To understand maths

What is the basis for reason? And mathematics?

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wtf
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Re: To understand maths

Post by wtf »

A_Seagull wrote: The theorems that are generated are then necessarily true within that system, for they have been generated following the axioms and inferential logic of the system. And only those theorems that it generates can be considered to be statements of mathematics.

It is the role of mathematicians to design the axioms from which the machine can be constructed.
Such a machine can not be constructed. That's what Turing established in the 1930's. This result is intimately related to the work of Gödel.

This article may be of interest. https://en.wikipedia.org/wiki/Entscheidungsproblem

From the article:
Wiki wrote: In mathematics and computer science, the Entscheidungsproblem (pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm], German for 'decision problem') is a challenge posed by David Hilbert in 1928. The Entscheidungsproblem asks for an algorithm that takes as input a statement of a first-order logic (possibly with a finite number of axioms beyond the usual axioms of first-order logic) and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

In 1936, Alonzo Church and Alan Turing published independent papers showing that a general solution to the Entscheidungsproblem is impossible, assuming that the intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing thesis.
There is no way to program a computer to recognize when a given statement is a theorem following from a given set of axioms. This is subtly different from what you suggested, namely generating all theorems. According to the article you linked,
Wiki wrote:...given unbounded resources, any valid formula can eventually be proven. However, invalid formulas (those that are not entailed by a given theory), cannot always be recognized.
https://en.wikipedia.org/wiki/Automated ... he_problem
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A_Seagull
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Re: To understand maths

Post by A_Seagull »

wtf wrote:
A_Seagull wrote: The theorems that are generated are then necessarily true within that system, for they have been generated following the axioms and inferential logic of the system. And only those theorems that it generates can be considered to be statements of mathematics.

It is the role of mathematicians to design the axioms from which the machine can be constructed.
Such a machine can not be constructed. That's what Turing established in the 1930's. This result is intimately related to the work of Gödel.

This article may be of interest. https://en.wikipedia.org/wiki/Entscheidungsproblem

From the article:
Wiki wrote: In mathematics and computer science, the Entscheidungsproblem (pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm], German for 'decision problem') is a challenge posed by David Hilbert in 1928. The Entscheidungsproblem asks for an algorithm that takes as input a statement of a first-order logic (possibly with a finite number of axioms beyond the usual axioms of first-order logic) and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

In 1936, Alonzo Church and Alan Turing published independent papers showing that a general solution to the Entscheidungsproblem is impossible, assuming that the intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing thesis.
There is no way to program a computer to recognize when a given statement is a theorem following from a given set of axioms. This is subtly different from what you suggested, namely generating all theorems. According to the article you linked,
Wiki wrote:...given unbounded resources, any valid formula can eventually be proven. However, invalid formulas (those that are not entailed by a given theory), cannot always be recognized.
https://en.wikipedia.org/wiki/Automated ... he_problem

I think we are talking at cross purposes here. I am not talking about a machine that will test a string of mathematical symbols to determine whether it can be generated from a finite set of axioms or not.

I am talking about a machine that is constructed according to a finite set of axioms which is then set in motion (and perhaps, but not necessarily, guided by human intervention) and which then generates strings of symbols, called theorems. That is all. I see no reason why such a machine cannot be constructed.
wtf
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Re: To understand maths

Post by wtf »

A_Seagull wrote: I am talking about a machine that is constructed according to a finite set of axioms which is then set in motion (and perhaps, but not necessarily, guided by human intervention) and which then generates strings of symbols, called theorems. That is all. I see no reason why such a machine cannot be constructed.
Of course we can build one right here. Here is the pseudocode.

For i = 1, 2, 3, ... print "i = i".

This machine prints the theorems 1 = 1, 2 = 2, 3 = 3, 4 = 4, ... and it never stops. Infinitely many true theorems!

That fits what you said but I don't think it's what you intend, unless you are satisfied by my revolutionary theorem generating machine. Can you be more specific? Because the trivial machine I just outlined satisfies your requirement. I'm assuming you mean (but are forgetting to explicitly mention) that you want all the true theorems, or only the true theorems, or only the interesting theorems, or all the truths of mathematics whether they're theorems or not. Each one of those ideas has difficulties. But it's easy to generate trivial theorems.
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A_Seagull
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Re: To understand maths

Post by A_Seagull »

wtf wrote:
A_Seagull wrote: I am talking about a machine that is constructed according to a finite set of axioms which is then set in motion (and perhaps, but not necessarily, guided by human intervention) and which then generates strings of symbols, called theorems. That is all. I see no reason why such a machine cannot be constructed.
Of course we can build one right here. Here is the pseudocode.

For i = 1, 2, 3, ... print "i = i".

This machine prints the theorems 1 = 1, 2 = 2, 3 = 3, 4 = 4, ... and it never stops. Infinitely many true theorems!

That fits what you said but I don't think it's what you intend, unless you are satisfied by my revolutionary theorem generating machine. Can you be more specific? Because the trivial machine I just outlined satisfies your requirement. I'm assuming you mean (but are forgetting to explicitly mention) that you want all the true theorems, or only the true theorems, or only the interesting theorems, or all the truths of mathematics whether they're theorems or not. Each one of those ideas has difficulties. But it's easy to generate trivial theorems.

Well, I think we are on the same page now. Your machine (call it wtf1) is a fine example of what I am talking about. Though I would add that its theorems are deductively true, albeit within the system wtf1, and are also inductively true in that they correspond with our general expectations of the real world.. ie A=A.

As you point out the wtf1 system is fairly trivial. But obviously more complex and interesting ones can be created.

And with regard to mathematics, the task is to create a machine that will generate the theorems that one is interested in generating, eg "9*9=81", "e**(i.pi)+1=0". The axioms that define the logic of the machine are then the axioms of the system. So far as the machine is concerned the strings of characters it generates as theorems are just meaningless strings of characters. yet they are 'true within the system as they have been proven to follow logically from the axioms.

It is then up to mathematicians, scientists and others to give them 'meaning' and to use them to model aspects of the real world.

And as you pointed out, such a system could never be 'complete' in that it would never generate every possible logical inference.

But there are many possible systems. The only limit on the axioms would be that they must be able to create a machine that can actually generate theorems.

The quality of a machine is then how 'interesting' are the theorems it can generate. eg Can they be used to map the real world ( eg complex numbers are useful for electronics), or how intrinsically interesting they are (such as Mandelbrot sets.)

How hard it might be to create a machine that would generate strings such as " there are an infinite number of prime numbers" or " the square root of 2 is irrational", I do not know. It might be expedient to have a second-order meta-system which can use the theorems of a first order system.

I dare say such a model for mathematics may not suit everyone. But it depends what one wants from a model. I certainly like its simplicity and it is sufficiently comprehensive for my needs.
wtf
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Re: To understand maths

Post by wtf »

I wonder if the OP got anything out of this discussion.

Automated theorem proving is interesting. Some mathematicians speculate that eventually computers will be capable of creative math. I don't believe that personally.
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A_Seagull
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Re: To understand maths

Post by A_Seagull »

wtf wrote:I wonder if the OP got anything out of this discussion.

Automated theorem proving is interesting. Some mathematicians speculate that eventually computers will be capable of creative math. I don't believe that personally.
Nor do I.
Dalek Prime
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Re: To understand maths

Post by Dalek Prime »

A_Seagull wrote:.
Are you Adam Seagull, from the COFR?
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A_Seagull
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Re: To understand maths

Post by A_Seagull »

Dalek Prime wrote:
A_Seagull wrote:.
Are you Adam Seagull, from the COFR?
No
Dalek Prime
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Re: To understand maths

Post by Dalek Prime »

A_Seagull wrote:
Dalek Prime wrote:
A_Seagull wrote:.
Are you Adam Seagull, from the COFR?
No
Thanks. Was curious, as his name was brought up yesterday in conversation, and I started wondering.
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NielsBohr
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Re: To understand maths

Post by NielsBohr »

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.
-If it can make you feel good, I already will tell you that most mathematicians are not interested in your deep questioning.

And not a single guy (since Goedel who took this option to speak to idiots) really think that a theory may be "complete". Depending on his ways (and on my sentence before), a theory can only be consistent.

But if maths were, there were not especially a "global understanding" (as you mention it) than a particular one. Anyway, first step is:
How do you define "understanding"?

In accordance with my logical redefinition (viewtopic.php?f=26&t=19705#p276215), mainstream logic even are in the impossibility to accord themselves to understanding.

As I show it, understanding requires time, because actual logic does.
Ed Dirac
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Re: To understand maths

Post by Ed Dirac »

Hi everyone.

I apologize for taking so long to answer. I haven't realize that there were answers. Thank you for answering and trying to clarify my doubts.
Impenitent wrote: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.

-Imp
But that only means that mathematicians make maths in certain language, It doesn't mean that maths are a language (I think).
Arising_uk wrote: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.
But we use logic to do mathematics, what is the difference between studying mathematics using some results to which we arrive using logic and studying mathematics logically? Anyway, your recommendations were helpful because both are of my interests.
Arising_uk wrote:The theorem-provers are written by humans it's that to actually work through some of the proofs could take forever if done by hand.
That’s very interesting, do you know some mathematical statement proved by these softwares?
wtf wrote:I hope I've at least explained what I mean when I say that set theory is the foundation of math
I’m sorry wtf, but I don’t understand what means that some theory or some collection of statements like set theory axiomatic system is or can be the foundations of mathematics. If you would want to explain it again I’d appreciate it very much. If you think it is already sufficiently well explained and there is no reason to explain it again, or simply you do not want to explain it again, no matter, I do not want to be tiresome.
wtf wrote:Set theory is arguably already losing its foundational importance. One alternative currently being developed is homotopy type theory, which is influenced by intuitionist philosophy as well as by computer science.
Do you know if homotopy theory can be a foundation of set theory at the same time that it is a foundation of mathematics?
wtf wrote: Some guy named Giuseppe Peano wrote down some axioms in 1989. The idea was in the air. This Wiki article gives some of the history.https://en.wikipedia.org/wiki/Peano_axioms.
The Wikipedia said about that that: “The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction.”
I suppose that here is referring to set fact, set statements like axioms. But for every axiom A we can set another collection of set axioms B from which A can be derived. So, do you know what means “to be a more basic fact than another”?
And why we need the Von Newman definition of natural numbers if what Peano wanted to do with his axioms was to axiomatize the natural numbers?
NielsBohr wrote:But if maths were, there were not especially a "global understanding" (as you mention it) than a particular one. Anyway, first step is:
How do you define "understanding"?
Sorry but English is not my mother tongue, what do you mean by “But if maths were, there were not especially a "global understanding” than a particular one”?
To know why what we say in maths is true and not false.
osgart
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Re: To understand maths

Post by osgart »

maybe math could become exact logic someday? And complete logic that can be applied exactly to any problem or actuality. To do that every symbol would have to have meaning that is accurate and exact. If the symbols are arbitrary and not exact measures or definitions it will never reflect reality actual. Math that way is a manipulation of reality and not a representation of it. How many straight up additions, subtractions, multiplications, and divisions are there in nature. But if each operator was like an exact verb or subject of an exact measure perhaps math could be strikingly accurate in real terms.
Say the exact formula for earth gravity was an operator than we could apply it 320 EG 103 = applied force of 320 to exact gravity on 103lbs at density=mass/volume over length of time. Than if each operation was dealt with in fragnitudes or partials, wholes, or magnitudes of accurate measure, positive or negative, than math could become exact logic language.
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