Re: Epistemology 201: The Science of Science

From: Jason (jasonstevensNOSPAM_at_free.net.nz)
Date: 02/03/05 

Date: Thu, 03 Feb 2005 13:34:53 +1300

>>>Keep in mind that mathematics is far older than Russell & Whitehead's
>>>Principia.
>
>
>>Yes, but most people have moved into modern times.
>
>
> Calculus, which is far older than Russell and Whitehead, is still the
> centerpiece of the undergraduate curriculum.

It has survived into modern times, unlike maths being an informal thing
that is not about the world.

> -------------
>
>
>>>The continuum hypothesis is a side show. For sure, some people find
>>>it an interesting side show. But it is still only a side show.
>
>
>>There are an infinite number of unprovable theorems. I wouldn't call
>>them a side-show. If people study maths for the sake of maths (divorsed
>
>>from the world) as you say, I also wouldn't call it a side show.
>
> You missed the point -- probably because I was not explicit.
>
> There isn't very much that depends on the continuum hypothesis. By
> contrast, if the Riemann hypothesis could be settled, that would have
> important consequences.

Not much depends on it? An infinite number of derivations depend on it.
  If something is not popular, it doesn't make it go away.

> -------------
>
>
>>>As a first approximation, think of mathematics as the study of
>>>pattern or regularity. When mathematicians come across an
>>>interesting pattern or regularity, they will attempt to characterize
>>>that regularity. This characterization might be in the form of a set
>>>of rules. Then they will use these rules as axioms, and investigate
>>>their consequences.
>
>
>>You just described many fields.
>
>
> Other fields may study particular observed patterns. Mathematics is
> more of a theory of patterns.

Ah, the measurable "more".

> -------------
>
>
>>>Axiom systems are not the starting point of mathematics. They are
>>>often one of the products of mathematics.
>
>
>>Maths can't produce its own axioms. It has access to statements and
>>logic so a formal system can be asserted in maths.
>
>
> The axioms used in mathematics all come from mathematics. The idea
> that mathematics cannot produce its own axioms is absurd.

Yes, see this is where we are refering to different things with the word
"mathematics". It is not so absurd when you consider maths as a formal
system, which is what I'm doing. I see that it is absurd to you because
you don't seem to follow where I'm coming from.

>>>It depends on what is being studied. For many purposes, you can get
>>>away with the axioms of a field. For other purposes, you will also
>>>need an axiomatization of topology.
>
>
>>Am I hearing "you need an axiomatization"? That is, you need the formal
>>system?
>
>
> Mathematics certainly works with axioms. We disagree because there
> isn't a single axiom system. There are many, and new ones are
> created as needed.

When something is proved, it is always with respect to an axiom system
(or a set of learned rules). If someone wanted to prove a theorem, they
don't just create a new axiom to do it, the 'standard' set of axioms (or
learned rules) are used.

> -------------
>
>
>>>>How about another then. It has been proven that in ZFC set theory, the formal
>>>>system of mathematics (I honestly can't see why you flatly refuse that there is
>>>>such a system), the continuum hypotheses is can neither be proven or disproven.
>>>>So it could be asserted true or false with a new axiom and there would be two
>>>>overlapping but distinct mathematical universes to choose from.
>
>
>>>Right. Should I add a yawn?
>
>
>>If you like, it seems to be how you approach the addition axiom. You
>>use it but don't acknowledge it.
>
>
> What, exactly, is this "addition" axiom?

Something that defines how "+" is used. It means something to us
because we have learned how to apply rules in its regard. But it is
just a mark on the page, so the addition axiom would look something like:

a1) (x + 0) => x
a2) (x + m) => (x + (m - 1)) + 1
a3) (x + 1) => x'

where ' is the successor.

>>>I'm not sure I would call it small, and I certainly would not say
>>>that it is inconsequential. Nevertheless, it is only part of
>>>mathematics. Do keep in mind that calculus is several hundred years
>>>older than ZFC.
>
>
>>Okay, I'm starting to get an appreciation of where you're coming from.
>>I think. As practitioners of maths you don't care about the engine of
>>maths until it breaks down.
>
>
> ZFC is not the engine of mathematics. I'm not sure that "the engine
> of mathematics" even makes sense.

It does to me. An "effective method" or "mechanical method" are
engine-like terms. Maths uses this in proofs.

> -------------
>
>
>> If this is fair, then it is a practical
>>consideration which is why I've taken a while to appreciate your point.
>> We're in a philosophy news group and philosophy typically deals with
>>impractical problems (bar applied phil I suppose).
>
>
> We are really in an AI newsgroup, and we are are way off topic. Hmm,
> this is widely crossposted, so people are coming at it from several
> different directions.

Oh I see! I'm from the sci.phil.meta. I didn't realise there was a
broadcast going down.

>
> -------------
>
>
>>>There is a sub-discipline of mathematics that studies foundations. And
>>>within that subdiscipline, people attempt to find out how much of
>>>mathematics can be built on ZFC as a foundation. That's a very interesting
>>>exercise. But it doesn't follow that mathematics really does depend on
>>>ZFC as a foundation.
>
>
>>>Here is an analogy. It can be shown that all logic operations are
>>>derivable from "not and" (what the NAND gate does). But it does not
>>>follow that all actual logic chips are built out of NAND gates.
>
>
>>Maths depends on something.
>
>
> Sure. It depends on the human mind, and on conventions that are
> tacitly shared between mathematicians.
>

Sure, social conventions. I think the argument is split between
theoretical maths and practical maths. We may well be crossing purposes.

Relevant Pages

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    ... >>Maths is an extension of FOPC, ... Mathematics is much older than FOPC, ... are the axioms of set theory, ... You are suggesting that maths is not this formal system, so I am lead to assume ...
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