To understand maths

What is the basis for reason? And mathematics?

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Arising_uk
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Re: To understand maths

Post by Arising_uk »

A_Seagull wrote:...

Why invoke set theory as foundational? - it just seems to introduce unnecessary complication. ...
I thought it a consequence of the history of mathematical development?
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!
I thought it was because they wanted to show that Mathematics has a solid foundation so built one from Logic?
It seems to me that all these attempts to create a formal axiomatic system for mathemnatics are trying to prove something from nothing. ...
Not nothing but Logic.
But of course you can't do that. ...
But they did?
So instead, the assumptions are hidden within their axiomatic assertions. ...
My thought is that your thoughts are because of the works of such as Russell and Whitehead and the consequent development of the field of axiomatic systems.
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.
Why?
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.
Syntactically but semantically?
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A_Seagull
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Re: To understand maths

Post by A_Seagull »

Arising_uk wrote:I thought it was because they wanted to show that Mathematics has a solid foundation so built one from Logic?
And just what is this 'logic' you refer to?
wtf
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Re: To understand maths

Post by wtf »

[Note: I wrote this long and ungainly post but Arising_uk said the exact same things only in way fewer words].

A_Seagull wrote: 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.
Perfectly ok by me. If you've gone from "No foundation exists," to "The standard foundation is not to my liking," you will get no argument from me and perhaps even some points of agreement.

I'm viewing math as a historically contingent human activity. Math was done for thousands of years without set theory. Archimedes and Euclid and Newton and Gauss and Euler never heard of the empty set. Yet they did math.

When I say that set theory is the foundation of math, that's a true statement about the world. I'm not saying that set theory is necessarily the foundation of math, or the only foundation or the best foundation or even a particularly good foundation; although it's had a successful century-long run. All I'm saying is that set theory is the foundation of math. We can like it or not but is a fact about the world.

As to how this happened, math had a so-called fondational crisis around the 1900's, the outcome of which was set theory. I wish I could say a lot more about this but in the interest of brevity I'll leave it to the Wiki page.

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. (The linked Wiki page isn't very helpful unfortunately). In fact one aspect of HoTT as they call it is computer verification of proofs. Clearly this will be a big thing in the future. You remember that detailed proof you challenged me to do. There's an app for that!

Another modern development is Category theory, which is based on philosophical structuralism. Objects are not characterized by what they are; but rather on their relationships with other objects. Most modern algebra and geometry is done in the framework of Category theory these days.

If you don't like the current state of things, that's perfectly fine. However if you want to know what the current state of foundations is, it's set theory. For no other reason than that's the way it turned out to be. Today. Next year, next decade, next century? Human thought moves ever forward.

A_Seagull wrote: Why invoke set theory as foundational? - it just seems to introduce unnecessary complication.
Lot of very good reasons. Goes back to the rigorization of calculus, the renewed interest in foundations after the discovery of non-Euclidean geometry, the discovery of set theory by Cantor, the discovery by Russell that Cantor's set theory was inconsistent, the growing need to get everything cleaned up. A lot of fascinating history and philosophy in there from say 1870 to 1940, say

A_Seagull wrote: 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.

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. Again it's a matter of historical contingency. Peano was born, became a math professor, people were trying to axiomatize the natural numbers, he worked on the problem. On some other planet they still have the natural numbers (presumably) but not the same set of axioms, if they even have axioms. Platonic math and human-made math, two different things. Except that maybe Platonic math doesn't exist and historically contingent math is all there is. Philosophers like to kick that one around.

A_Seagull wrote: 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 was in the air. There were all these mathematical and philosophical problems around. Russell went after the problem. It's a singular work, very difficult. I have no idea why Russell did what he did.

But I will disagree with you about making 1 + 1 = 2 an assumption. You haven't told me what the symbols mean. And around 1900 it became really important to get things nailed down to make sure mathematical reasoninig was valid. Nobody knew what the right principles were. It was all part of a long historical process.

Remember that during this time, relativity and the quantum theory were being discovered. You could ask why those theories popped up and why they took the form that they did. All good questions in the history of thought.

A_Seagull wrote: 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.
The assumptions aren't hidden. For the first time they are clearly revealed. Now we know how to make sense of the infinitary operations of calculus. We know what are the underlying principles of number theory. We understand the power and limitations of mathematical logic. All this had to be worked out explicitly by smart but fallible people over many decades.

A_Seagull wrote: 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.
Ah the axioms are NOT complete! That's what Gödel showed in the 1930's: that no matter what axiom system you pick, there will always be mathematical facts that are true but that can't be proved in that system of axioms. This was a big deal, one of the profound intellectual shocks of the twentieth century. And Gödel's work came directly out of Russell's. In fact Gödel was working in Russell's system to show that axiomatic systems were inherently incomplete.

A_Seagull wrote: 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.
If you like the relationship-oriented approach, structuralism and Category theory are for you. https://en.wikipedia.org/wiki/Structura ... thematics) [link doesn't catch closing paren, need to copy/paste]

I hope I've at least explained what I mean when I say that set theory is the foundation of math; even if it's not philosophically satisfying to you. It's just the way it is. Everyone who studies math works in the framework of set theory. In the future, who knows. It'll all be done by computer, even the proofs. The state of math is always changing because math is a historically contingent activity of humans.
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Arising_uk
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Re: To understand maths

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A_Seagull wrote:And just what is this 'logic' you refer to?
In what sense?
Impenitent
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Re: To understand maths

Post by Impenitent »

...
wtf wrote:I hope I've at least explained what I mean when I say that set theory is the foundation of math; even if it's not philosophically satisfying to you. It's just the way it is. Everyone who studies math works in the framework of set theory. In the future, who knows. It'll all be done by computer, even the proofs. The state of math is always changing because math is a historically contingent activity of humans.
done by computer...

thus ending history?

programmed by humans or will the computers program themselves?

-Imp
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Arising_uk
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Re: To understand maths

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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.
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A_Seagull
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Re: To understand maths

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wtf wrote: But I will disagree with you about making 1 + 1 = 2 an assumption. You haven't told me what the symbols mean.

Perhaps the symbols don't 'mean' anything. They are just symbols. If you (or I ) choose to associate those symbols with, say, sheep in a field, then we may do so. But otherwise the symbols are entirely abstract.


A_Seagull wrote: 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.
Ah the axioms are NOT complete! That's what Gödel showed in the 1930's: that no matter what axiom system you pick, there will always be mathematical facts that are true but that can't be proved in that system of axioms. This was a big deal, one of the profound intellectual shocks of the twentieth century. And Gödel's work came directly out of Russell's. In fact Gödel was working in Russell's system to show that axiomatic systems were inherently incomplete.

But there were pre-assumptions in Godels work. Why assume that every statement in mathematics can be labelled as either true or false? What is meant by a 'mathematical statement'? What is meant by 'true' in this instance? While not wanting to belittle Godels work, I do question its relevance.

I think it depends upon what one wants from mathematical foundations. And that is where philosophy becomes relevant.

All I want from the foundations of mathematics is an efficient set of axioms that can generate the theorems of mathematics. The symbols are entirely abstract. The totality of the system is defined by the system itself. The theorems that it generates are by definition 'true' albeit only within that system.

The system would be infinite in terms of the number of theorems it can generate, so there would be no question about 'completeness'.
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Arising_uk
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Re: To understand maths

Post by Arising_uk »

A_Seagull wrote:...
The system would be infinite in terms of the number of theorems it can generate, so there would be no question about 'completeness'.
Not sure how it works in maths but how does infinity affect 'completeness'?

As in Logic I thought it just that every formula could be derived from the system but not time limited?

As an aside, Russell thought Godel misunderstood him, and whilst I'm not a mathematician and what I've read from the philosophical point of view was way above my comprehension, I did think that what Godel may have just shown is that Maths and Logic are not equivalents, as from the little I understood Russel 'logicised' Maths and then Godel 'mathematised' Logic to produce formulas that could not be proved in the logic under question but could this not just mean that the operations of Maths are different from the operation of Logic? Just asking.
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Re: To understand maths

Post by Impenitent »

a self referential system ... proven and justified through circular reasoning

Ludwig was here

-Imp
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Re: To understand maths

Post by Necromancer »

Impenitent wrote:a self referential system ... proven and justified through circular reasoning

Ludwig was here

-Imp
https://en.wikipedia.org/wiki/Circular_reasoning Maybe the premises are not the conclusion? :wink:
Last edited by Necromancer on Thu Jul 14, 2016 1:51 am, edited 2 times in total.
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Re: To understand maths

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Impenitent wrote:a self referential system ... proven and justified through circular reasoning

Ludwig was here

-Imp
And showed it not circular but grounded upon things and states of affairs.
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Re: To understand maths

Post by Impenitent »

self referential systems are closed...

yet "grounded" through induction?

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Re: To understand maths

Post by wtf »

A_Seagull wrote: Perhaps the symbols don't 'mean' anything. They are just symbols. If you (or I ) choose to associate those symbols with, say, sheep in a field, then we may do so. But otherwise the symbols are entirely abstract.
All the more reason to get incredibly precise about what the rules are for manipulating symbols! That's what the foundational crisis was all about. That's exactly what Cantor, Russell, Frege, Zermelo, and many other brilliant mathematicians, logicians, and philosophers were doing. Working out the precise rules for manipulating symbols to do mathematics.
A_Seagull wrote: But there were pre-assumptions in Godels work. Why assume that every statement in mathematics can be labelled as either true or false? What is meant by a 'mathematical statement'? What is meant by 'true' in this instance? While not wanting to belittle Godels work, I do question its relevance.
Of course those are great questions. Those are exactly the questions mathematicians were asking, and answering. By the time Gödel came around in the 1930's, he had all that work to build on.

I'm not sure how you can question its relevance. At the exact same time, Turing came up with his theory of computation, which has many points of connection with Gödel's incompleteness theorem. And one can hardly claim that the theory of computation is not relevant. On the contrary, we live in Turing's world. Which means, we live in Gödel''s world as well.
A_Seagull wrote: I think it depends upon what one wants from mathematical foundations. And that is where philosophy becomes relevant.


Of course. That's EXACTLY what the mathematical philosophers were working out during this period of time. It seems to me that the work being done in foundations during that period of time is exactly what you want! They were doing all this stuff. They asked those questions and they came up with answers, which are with us today.
A_Seagull wrote: All I want from the foundations of mathematics is an efficient set of axioms that can generate the theorems of mathematics. The symbols are entirely abstract. The totality of the system is defined by the system itself. The theorems that it generates are by definition 'true' albeit only within that system.
That's exactly what set theory does. One can be a Platonist and think the symbols mean something; or a formalist and treat it all like an abstract game. This is exactly what came out of the foundational crisis. This is exactly what all this work from 1870 through 1940 or so was about.
A_Seagull wrote: The system would be infinite in terms of the number of theorems it can generate, so there would be no question about 'completeness'.
There are infinitely many theorems. But Gödel proved that in any system of axioms rich enough to model number theory, there are statements that can neither be proved nor disproved from those axioms. The proof has been checked and rechecked for 80 years. It's true.

This is a profound result. Gödel's work is simply a fact of modern intellectual life. Perhaps an analogy is relativity, or the quantum theory, both of which were being developed at exactly the same period of history.
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Re: To understand maths

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Arising_uk wrote:Not sure how it works in maths but how does infinity affect 'completeness'?
You get incompleteness in any axiomatic theory strong enough to express number theory. By number theory is typically meant a Peano-like structure with induction. Like the natural numbers. Once you believe in the set of natural numbers, you have infinity in math and everything else follows.

Of course there's no incompleteness in a finite universe, but there are no natural numbers either. At least not all of them.
Arising_uk wrote: As in Logic I thought it just that every formula could be derived from the system but not time limited?
Time has nothing to do with any of this. Proofs from axioms are just finite sequences (proofs) of finite strings of symbols (axioms and intermediate theorems).
Arising_uk wrote: As an aside, Russell thought Godel misunderstood him ...
I find that very funny! Of course it makes perfect sense. Russell was old school, Gödel's work came much later. It's natural for an old expert to not believe in the work of his own young disciples. There's a quote about science advancing not because the new theories convince everyone; but because the believers in the old theories die off. Lot of truth to that.

Arising_uk wrote: , and whilst I'm not a mathematician and what I've read from the philosophical point of view was way above my comprehension, I did think that what Godel may have just shown is that Maths and Logic are not equivalents,
Yes exactly! One early philosophical hope was that math would turn out to be logic. I believe that's what Russell was trying to do. Gödel destroyed that hope forever.

Arising_uk wrote: as from the little I understood Russel 'logicised' Maths and then Godel 'mathematised' Logic to produce formulas that could not be proved in the logic under question but could this not just mean that the operations of Maths are different from the operation of Logic? Just asking.
That's my understanding. Math isn't logic.
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Re: To understand maths

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wtf wrote:
A_Seagull wrote: I think it depends upon what one wants from mathematical foundations. And that is where philosophy becomes relevant.


Of course. That's EXACTLY what the mathematical philosophers were working out during this period of time. It seems to me that the work being done in foundations during that period of time is exactly what you want! They were doing all this stuff. They asked those questions and they came up with answers, which are with us today.
story.
I suspect that what people want from mathematical foundations will depend upon the general philosophy with which they want it to be interfaced.

There would certainly seem to be no problems with mathematics itself, it all seems to be fine with no outstanding inconsistencies.

All I want for a model is for it to show how mathematics can be used for science and every day life.

So the model that achieves this is one whereby the axioms of mathematics are used to construct a logical machine that can generate the theorems of maths. (This could be an idealised machine in the form of a Turing machine, or perhaps a real one in the form of a computer.)

The best reference I can find for this is: https://en.wikipedia.org/wiki/Automated_theorem_proving . But I am talking about a machine that is capable of generating more basic theorems such as 3+4=7 , e^{ipi }+1=0...... as well as more complex ones.

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. It is the role of scientists (and others) to tie or map the theorems to aspects or measurements of the real world.

So far as I can tell, that is all that is required for a basic model for the foundations of mathematics and for its interface with other branches of knowledge.
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