Mathematical Knowledge

[Master Lawbringer]

Just testing. This used to be uncontroversial :

All knowledge is ultimately circular. Break any idea down long enough and you'll end up with ideas, like 'time', for which all definitions end up circular.

Specifically concerning numbers : You can't escape the fact that trying to define what a number actually is begins and ends with the pragmatic observation that we, and other machines, are able to count. Logic and set theory, themselves based on self-evident, circular, concepts (try to define 'set') are circularly dependent on each other and even if you reduce everything to just manipulations of symbols you'll just end up with a machine that can count and perform calculations.

You can't escape the self-evident and circular nature of the fundamental ideas.

Y'all probably disagree with this but tell me why ...

[studiot]

26 minutes ago, Master Lawbringer said:

You can't escape the fact that trying to define what a number actually is begins and ends with the pragmatic observation that we, and other machines, are able to count. Logic and set theory, themselves based on self-evident, circular, concepts (try to define 'set') are circularly dependent on each other and even if you reduce everything to just manipulations of symbols you'll just end up with a machine that can count and perform calculations.

Mathematics is not only about numbers and counting and fraction of maths taken up with this is diminishing.

You do not need set theory or to be able to count to define numbers, though both are useful in keeping things tidy

Number theory itself extends more widely than the counting numbers.

 

Edit : I now see that you are reintroducing your very recent, more jumbled, topic placed in the trash can by the moderators. Hopefully you have cleared this simpler version with them?

Edited April 2, 2020 by studiot

[Master Lawbringer]

12 minutes ago, studiot said:

Mathematics is not only about numbers and counting and fraction of maths taken up with this is diminishing.

You do not need set theory or to be able to count to define numbers, though both are useful in keeping things tidy

Number theory itself extends more widely than the counting numbers.

Are you claiming that there are areas of mathematics for which the following does not hold :

You can't escape the self-evident and circular nature of the fundamental ideas.

???

[studiot]

36 minutes ago, Master Lawbringer said:

You can't escape the self-evident and circular nature of the fundamental ideas.

 

I note your statement could itself be considered 'circular' since it uses a mathematical notion (circular) to  describe something 'more fundamental'.

What do you think 'the fundamental ideas' of mathematics are?

Surely in any system of mathematics you introduce 'fundamental ideas' , often called axioms, and the important thing is that they are consistent with each other.
Where is the circularity in that?

Edited April 2, 2020 by studiot

[Master Lawbringer]

1 hour ago, studiot said:

 

I note your statement could itself be considered 'circular' since it uses a mathematical notion (circular) to  describe something 'more fundamental'.

What do you think 'the fundamental ideas' of mathematics are?

Surely in any system of mathematics you introduce 'fundamental ideas' , often called axioms, and the important thing is that they are consistent with each other.
Where is the circularity in that?

Go on in that direction and you'll end up with statements like 'The reason cannot describe itself' which is a Liar Paradox. And then it's just boom ... absurdity.
Observing this you can make the decision to either disregard this and stay within the confines of the reason or you could observe that it's just always like this : This explosion into absurdity is just a hint towards the deeper truth behind statements like 'The reason cannot describe itself.'
Obviously, I chose the latter option, but this does not mean that I think that reason and mathematics aren't pragmatically useful.

Also pragmatically I consider the ability to count as a starting point for mathematics. The concepts that appear then, the numbers, also turn out to be irreducible anyway.

Just because the system is consistent, which basically means that it's not logically trivialistic, doesn't explode into absurdity, doesn't mean that the base concepts aren't self-evident and circular notions. And circularity appears the whole time. Rewriting equations is basically circular logic which shows that such logic isn't actually fallicious, unless you mean that a circular definition adds actual new knowledge that the original statement did not contain.

[Strange]

2 hours ago, studiot said:

Edit : I now see that you are reintroducing your very recent, more jumbled, topic placed in the trash can by the moderators. Hopefully you have cleared this simpler version with them?

!

Moderator Note

I am happy with it so far. 

It introduces, reasonably clearly, an important fact about definitions which could lead to an interesting discussion. I will move it to Philosophy though, as the issue is wider than mathematics. (I will also be keeping a close eye on it.)

 

[taeto]

2 hours ago, Master Lawbringer said:

Y'all probably disagree with this but tell me why ...

What is wrong with solving problems by computation? After all it is what people tend to do these days.

Maybe it also helps to know if and where you have possibly seen an actual mathematical definition that is circular?

Edited April 2, 2020 by taeto

[studiot]

32 minutes ago, taeto said:

Maybe it also helps to know if and where you have possibly seen an actual mathematical definition that is circular?

Excellent question from taeto there. +1

 

59 minutes ago, Master Lawbringer said:

Also pragmatically I consider the ability to count as a starting point for mathematics. The concepts that appear then, the numbers, also turn out to be irreducible anyway.

Thank you for replying to my question.

However I must disagree with your answer.

There is a more fundamental process in Mathematics than counting. As evidenced by relics that archaeologists have discovered this goes at least as far back as the stone age.
It was also in use into the 20th century, as evidenced by instructions to land surveyors performing chain surveys.

 

I hinted at this in my first post.

3 hours ago, studiot said:

You do not need set theory or to be able to count to define numbers

 

 

[Master Lawbringer]

47 minutes ago, taeto said:

What is wrong with solving problems by computation? After all it is what people tend to do these days.

Maybe it also helps to know if and where you have possibly seen an actual mathematical definition that is circular?

http://geocalc.clas.asu.edu/GA_Primer/GA_Primer/introduction-to-geometric/defining-and-interpreting.html

The first thing that came to my mind is how the geometric product is defined in geometric algebra. The inner and outer product are defined using the geometric product, and the geometric product is defined using the inner and outer product. Just look closely.

[taeto]

24 minutes ago, Master Lawbringer said:

http://geocalc.clas.asu.edu/GA_Primer/GA_Primer/introduction-to-geometric/defining-and-interpreting.html

The first thing that came to my mind is how the geometric product is defined in geometric algebra. The inner and outer product are defined using the geometric product, and the geometric product is defined using the inner and outer product. Just look closely.

Thank you for giving an answer +1.

But the page that you linked does not define the geometric product using the inner or outer product, so far as I can see, since the four listed properties of the geometric product do not contain any expression with a \(\cdot\) in it. Can you point to the line of text where you think this happens? 

I would agree if you argue that the definition is incorrect. It allows \( a\cdot b = 0\) for all vectors \(a,b,\) which is likely not intended. Even that does not make it circular.

Edit: Forget the last paragraph, I missed that it says that \(|a|\) must be positive, so nonzero. So the thing is only incorrect in saying that \(|0|\) has to be positive.

Edited April 2, 2020 by taeto

[studiot]

16 minutes ago, Master Lawbringer said:

http://geocalc.clas.asu.edu/GA_Primer/GA_Primer/introduction-to-geometric/defining-and-interpreting.html

The first thing that came to my mind is how the geometric product is defined in geometric algebra. The inner and outer product are defined using the geometric product, and the geometric product is defined using the inner and outer product. Just look closely.

I don't see any relevence to anything in your opening assertions about counting here.

 

You offered a link to a particularly elementary introduction to linear algebra specifically stated to be for non mathematicians, quoting results that are not part of the axioms of a linear space.

How are the standard axioms of linear algebra and vector spaces circular ?

 

Edited April 2, 2020 by studiot

[Master Lawbringer]

12 minutes ago, taeto said:

Thank you for giving an answer +1.

But the page that you linked does not define the geometric product using the inner or outer product, so far as I can see, since the four listed properties of the geometric product do not contain any expression with a ⋅ in it. Can you point to the line of text where you think this happens? 

FROM THE WEBSITE :

In terms of the geometric product ab we can define two other products, a symmetric inner product

          (1)    a∙b = ½(ab + ba) = b∙a

and an antisymmetric outer product 

          (2)    a∧b = ½(ab − ba) = − b∧a

Adding (1) and (2), we obtain the fundamental formula

          (3)    ab = a∙b + a∧b called the expanded form for the geometric product.

END FROM THE WEBSITE.

In the development of a mathematical system the circularity becomes obfuscated. Do the first set of rules imply the (circular) definition of the geometric product in terms of the inner and outer product or is it the other way round? And this is irrelevant as long as the system is consistent.

Complexity doesn't make this circularity disappear. On analysis, it's always there due to the circular nature of the base concepts.

[taeto]

5 minutes ago, Master Lawbringer said:

FROM THE WEBSITE :

In terms of the geometric product ab we can define two other products, a symmetric inner product

          (1)    a∙b = ½(ab + ba) = b∙a

and an antisymmetric outer product 

          (2)    a∧b = ½(ab − ba) = − b∧a

Adding (1) and (2), we obtain the fundamental formula

          (3)    ab = a∙b + a∧b called the expanded form for the geometric product.

END FROM THE WEBSITE.

In the development of a mathematical system the circularity becomes obfuscated. Do the first set of rules imply the (circular) definition of the geometric product in terms of the inner and outer product or is it the other way round? And this is irrelevant as long as the system is consistent.

Complexity doesn't make this circularity disappear. On analysis, it's always there due to the circular nature of the base concepts.

I am pretty sure you are reading it wrong. The formula (3) is not a definition of anything; it is an identity that follows from the previous definitions of geometric and inner and outer products. It is similar to first defining what you mean by '2', '+' and '4', and then deriving the identity '2+2=4' from those definitions. In fact adding (3) as a definition would not make anything circular anyway, since it is just an easy consequence of the previous conditions given for the products. 

[Master Lawbringer]

How would you define '2'? First there is the pragmatic observation that we can count. Why can we count? Well ... pragmatically I observe that I'm just able to do such a thing. So that would become circular if I tried to define it in that way. However we also observe that numbers can be defined in terms of other numbers and in this way it appears, at first, to be closed in itself. Is it completely closed in itself? Enter the Foundational Crisis ...

[taeto]

Since you started out by mentioning how numbers are defined, I assume that you are already familiar with Peano's axioms:

en.wikipedia.org/wiki/Peano_axioms

I also assume that you now agree that nothing is circular in your geometric example.

[Master Lawbringer]

1 minute ago, taeto said:

Since you started out by mentioning how numbers are defined, I assume that you are already familiar with Peano's axioms:

en.wikipedia.org/wiki/Peano_axioms

I also assume that you now agree that nothing is circular in your geometric example.

I already wrote this and it also concerns Peano's axioms :

 Logic and set theory, themselves based on self-evident, circular, concepts (try to define 'set') are circularly dependent on each other and even if you reduce everything to just manipulations of symbols you'll just end up with a machine that can count and perform calculations.

You can't reduce numbers to something simpler. You might reformulate them and that can be useful pragmatically but there's no real reduction going on, in the philosophical sense.

And no, I most certainly do not agree that nothing is circular. The only thing that could make it non-circular is Hilbert's Program and that also failed.

[taeto]

7 minutes ago, Master Lawbringer said:

I already wrote this and it also concerns Peano's axioms :

 Logic and set theory, themselves based on self-evident, circular, concepts (try to define 'set') are circularly dependent on each other and even if you reduce everything to just manipulations of symbols you'll just end up with a machine that can count and perform calculations.

You can't reduce numbers to something simpler. You might reformulate them and that can be useful pragmatically but there's no real reduction going on, in the philosophical sense.

And no, I most certainly do not agree that nothing is circular. The only thing that could make it non-circular is Hilbert's Program and that also failed.

If your notion of `circular' only means that in the definition of one concept there is reference to another concept, then your notion differs from the usual understanding. You can choose to stick with your own private understanding. But it will make communication difficult.

[Master Lawbringer]

I mean that if you perform an exhaustive analysis, like reducing any concept to more basic concepts and then see where you end up, the circularity becomes apparent. The ride on the merry go round of rationality might be complicated to the point where its essential circular nature is obfuscated if you're only willing to look at a part of the path and disregard the totality.

[dimreepr]

10 minutes ago, Master Lawbringer said:

I mean that if you perform an exhaustive analysis, like reducing any concept to more basic concepts and then see where you end up, the circularity becomes apparent. The ride on the merry go round of rationality might be complicated to the point where its essential circular nature is obfuscated if you're only willing to look at a part of the path and disregard the totality.

I'm not sure you understand the point of mathematics. 

 

A square is axiomatic, how is that circular???

[Master Lawbringer]

And the reason I consider numbers fundamental can not be anything else (here) than merely the pragmatic observation on their ubiquitous and irreducible nature.

[Strange]

I know I am not supposed to get involved as I have already acted as a moderator (but hopefully in a fairly non-contentious way), but I can't resist the need to point out that there are two ways of defining something:

1. In terms of something else (which is, hopefully, simpler)

2. As an axiom (or postulate or "self evident truth" or whatever) in which case the thing is defined as simply being itself. (I believe this is what "Lawbringer" is referring to as a "circular definition".)

All definitions must eventually bottom out to (2).

I would not call that a circular definition as I think it is useful to distinguish fundamental concepts that cannot be defined in terms of anything else, from the circular definitions which are often the basis of a fallacy (similar to begging the question). 

[studiot]

31 minutes ago, dimreepr said:

A square is axiomatic, how is that circular???

  +1

 

18 minutes ago, Master Lawbringer said:

And the reason I consider numbers fundamental can not be anything else (here) than merely the pragmatic observation on their ubiquitous and irreducible nature.

 

You have said that before in this thread and already been told you are not correct, and even given examples.

 

It is evident that you do not wish to address anything that disagrees with your unsupported viewpoint.

 

 

[Master Lawbringer]

I already said that circular reasoning is only fallacious if it is used to imply that new information is added.
And you can see that a concept must be assumed to be self-evident when it turns out to be directly circular. Maybe I should have said directly circular?
Like 'time'. Just try it. You just get 'time' is ... a 'period' ... and a 'period' is ... a time. When some process like that occurs you know you've hit rock bottom.
(Ah, but we _can_ measure it. Which takes me to point 2. of my original document but ... endless debate ...)

10 minutes ago, studiot said:

You have said that before in this thread and already been told you are not correct, and even given examples.

It is evident that you do not wish to address anything that disagrees with your unsupported viewpoint.

Are you saying that numbers aren't ubiquitous in our lives or are you still claiming that they can be reduced to simpler things in the philosophical sense?

And you don't want to hear what I have to say about Bell's Theorem, so, there.

[studiot]

15 minutes ago, Master Lawbringer said:

I already said that circular reasoning is only fallacious if it is used to imply that new information is added.
And you can see that a concept must be assumed to be self-evident when it turns out to be directly circular. Maybe I should have said directly circular?
Like 'time'. Just try it. You just get 'time' is ... a 'period' ... and a 'period' is ... a time. When some process like that occurs you know you've hit rock bottom.
(Ah, but we _can_ measure it. Which takes me to point 2. of my original document but ... endless debate ...)

Are you saying that numbers aren't ubiquitous in our lives or are you still claiming that they can be reduced to simpler things in the philosophical sense?

And you don't want to hear what I have to say about Bell's Theorem, so, there.

I didn't make any such claim.

I said that numbers are not fundamental to Mathematics.

We now know plenty of mathematics that the concept of number doesn't come into.

Nor did I say that they can be deduced from anything simpler (although in fact there is a route to numbers as well as some set theory from simpler and more fundamental ideas)

What do you think humankind did before numbers were invented/discovered ?

 

Here is a number system, adequate for the people that invented it,  (though I would guess that most if not all Australian Aboriginals can now use our modern one).

One ; Two : Many.

But there were simpler systems still.

From your attidude I am getting the vibes that you do not wish to know about these.

I already referred to these neolithic systems.

 

 

[Master Lawbringer]

Actually, one story I always enjoyed is the story about how ancient sheep herders used to know when they didn't have all the sheep at the end of the day, even though their mathematical abilities were similarly basic. They just used rocks. One rock for each sheep. And if there were rocks left at the end of the day they knew, even though they couldn't count that far, that they had missing sheep. The rocks were called calculi and that's where the word calculus comes from.
I guess humankind was more pragmatic in those times.
Similarly Cantor used this one to one correspondance idea for his proofs on infinities. You can't actually count to infinity, now, can you? But if you know that there's a one to one correspondance ...