Mathematical Knowledge

[studiot]

1 hour ago, Master Lawbringer said:

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 ...

Wa..al thaar ya go!

You knew about it all along.

I hadn't heard (pun intended) about the shepherd's story, but it makes sense. +1
That must be the explanation for all those piles of old stones knocking about on Salisbury Plain. Now I know why they built Stonehenge.

My neolithic reference was actually to what are called 'tally sticks'.
These have been found and studied and performed a similar function.

 

And yes, one-one correspondence is an incredibly powerful mathematical tool.

Not only does it underlie set theory, both finite and infinite, but it also underlies other parts of maths.
For example similarity transformations.
And shape in general. It allows you to see that apentagon is not a hexagon, for instance.

Shape is a fundamental mathematical notion, that exists quite independently of any number system or measure.

[taeto]

On 4/2/2020 at 4:21 PM, Master Lawbringer said:

I already said that circular reasoning is only fallacious if it is used to imply that new information is added.

I appreciate that, and agree with what you say here in your quote. 

It still irks me a little when you suggest that certain mathematical definitions are circular, when they are clearly not. It is kind of a tautology, since if any text purports to be a mathematical definition, and it is in ordinary use 'circular', then it could not be a mathematical definition at all. I still do not have an idea why you would suggest that the example of 'geometrical product' can be considered circular. But I suspect that a certain fact about it may present a clue. If I am right, then the word you are looking for in place of 'circular' might be 'incomplete'. 

I know a few things about how mathematics works, and I am very much a layperson when it comes to philosophy. I can well stand to be corrected as I proceed to say something 'philosophical' as follows. First, I suspect that you expect mathematical definitions to be largely of a different nature than definitions used in the outer world.

I have in mind something like the following definition of 'a cat': It is something which (i) is an animal, (ii) has four legs, and (iii) has a tail. In the outer world it serves as a description of an object. In mathematics it would be a definition of the term 'cat', providing one is already familiar with the terms 'animal', 'leg', 'tail', as well as with the number four, as mathematical terms.

For an example of an outer world cat, you may consult youtube, say by searching for 'lolcats'. For an example of a mathematical cat, we could let 'animal' be defined as an abstract group, a 'leg' an element of the group, and a 'tail' a neutral element of the group. In which case \(\mathbb{Z}_4,\) the cyclic group of order four, and the Klein \( 4\) -group \(\mathbb{Z}_2\times \mathbb{Z}_2\) are two examples of such mathematical cats.

What is 'wrong' then with the definition (i)-(iii) in the outer world case, and in the mathematical case? 

Outer world case:

We can assume, I hope, that the concepts of animals, legs and tails are already well-known, I hope, e.g. even chairs may have legs, and kites may have tails, but the meaning of the terms is clear.

(a) there are objects (e.g. dogs) that satisfy the definition, yet they are not cats (\(\alpha\) error), or 

(b) it does not correctly capture the meaning of 'cat', since some actual cats are without tails (\(\beta\) error), or

(c) it does not define any particular cat (\(\gamma\) error). such as my own pet cat, hence the definition is not complete, meaning that it does not define exactly that which I am interested in.

Mathematical case:

Again, the meaning of 'animal'=group and 'leg'=element and 'tail'=neutral element are assumed  already known. We notice that outer world words are sometimes used for mathematical objects, without the meaning carrying over much.

(a) An \(\alpha\) error does not occur, since every object that satisfies the definition of a 'cat', such as a dog, is a cat by definition. A dog may be called a dog-type cat, that is all.

(b) A \(\beta\) error does not occur, because an object which does not satisfy the definition of a 'cat' is not a cat, by definition.

(c)  There is \(\gamma\) error, since there exist more than one (in fact exactly two) cat, which makes the definition of a cat incomplete.

In conclusion, you seem to consider the case of \(\gamma\) error to qualify as 'circular'. 

But let me check again, because you told me to "define '2'". That was in the context of natural numbers, and so the request makes sense. The character/symbol '2' is used as a name for various objects in mathematics, such as the natural number 2, the rational number 2, the real number 2, the complex number 2, etc. I can offer this explanation of '2' in Peano Arithmetic: "the number two is the successor of the successor of the number zero". Does that seem to your satisfaction? 

 

Edited April 4, 2020 by taeto