Re: Epistemology 201: The Science of Science

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

Date: Thu, 03 Feb 2005 08:19:32 +1300

>>>>Maths is an extension of FOPC, like PA.
>
>
>>>Not really. Mathematics is much older than FOPC, so it doesn't make
>>>sense to say it is an extension of FOPC.
>
>
>>Okay, this is really strange to me because this is so not what I've come to
>>understand mathematics as. These days, in mathematical reasoning, logical
>>arguments are used to deduce consequences (theorems) of the assumptions of maths
>>(axioms). Most of maths is built from sets, so the basic assumptions of maths
>>are the axioms of set theory, in particular ZFC set theory. [Chapter Zero -
>>Fundamental Notions of Abstract Mathematics, Carol Schumacher]
>
>
> You appear to be thinking of mathematics as a branch of logic. But
> it is the other way around -- logic is a branch of mathematics.

Nah, logic is a subset of mathematical logic. But I guess you have just
as much right to decide what "mathematics" refers to as anybody.

>>You are suggesting that maths is not this formal system, so I am lead to assume
>>that you have some sort of prior understanding of what is mathematically legal
>>and illegal, like most people.
>
>
> "Mathematically legal" is a matter of logic. But mathematics is not
> just logic.

Correctish. Mathematically legal is also not just a matter of logic.

>> But is this type of reasoning informal or have
>>we our own set of assumptions, much like axioms, that enable us to perform
>>mathematical inference. When there is a disagreement, where do we turn? From
>>my understanding it is this formalised system of mathematics, which took root
>>with Whitehead and Russell in the principia mathematica. Hence its FOPC roots.
>
>
> Keep in mind that mathematics is far older than Russell & Whitehead's
> Principia.

Yes, but most people have moved into modern times. I respect that you
consider maths historically though.

>>I can accept that the axioms are not often invoked in the heat of proofs, but
>>then neither is the road-code when we are driving. Axioms as such don't need to
>>be the way to go either. The more intuitive way to go are to use rules of
>>inference, which are equivalent and perhaps closer to the story about how we
>>'do' maths.
>
>
>>Out of interest, if maths is not this formal system then how can abstract
>>mathematics take place? For example, how can the continuum hypothesis be
>>(dis)proven, or proved not to be provable?
>
>
> 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.

> 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.

> 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.

>>>>and assumed, as far as I am aware.
>
>
>>>Again, not really. Mathematicians often try to make do with minimal
>>>axioms.
>
>
>>Which ones? The choice is critical to what is provable and what isn't.
>
>
> 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?

>>>If you happen to be making a vague reference to the Banach-Tarski
>>>paradox, then you have it wrong. Banach-Tarski does depend on the
>>>axiom of choice.
>
>
>>I went to a seminar on this last year and I thought the dude said the problem
>>went away with invoking the axiom of choice. But now having read some more I
>>realise I misunderstood. Okay, bad example.
>
>
>>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.

>>If ZFC is assumed as the foundations of maths, it has been shown by Chaitin that
>>there are infinite arithmetic truths that cannot be proven in ZFC. Where does
>>maths as not-a-formal-system fit into this?
>
>
> Mathematics can manage pretty well without a foundation. Foundations come
> and go, but most of mathematics continues unperturbed.

Sure, its foundations can be considered descriptive, but then they
wouldn't be foundations would they. The foundations or infrastructure
of maths can change without effecting most people, but this is a
practical effect. In theory it is critical to what can be proved and
what can't be.

>>>There you go again. You talk about "the formal system of maths", but there
>>>is no such formal system. Then you suggest that we should instead
>>>study some other formal system. It is gibberish.
>
>
>>Is ZFC set theory a small and inconsequential part of mathematics?
>
>
> 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. 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).

>> I suppose
>>you don't really get into it unless you study number theory, mathematical logic
>>and stuff, but it was my understanding that this system what the foundation of
>>modern maths.
>
>
> It isn't the foundation of mathematics.
>
> 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. If I were to say "2 / 23 * =", you would
say... "what?" If I were to say "2 + 3 = 2", you would say "bollix."
How do we know this? We have learned, like riding a bike, the language
and rules of maths. Some of us might use nand gates to do this and
others might use nor gates, in logic it doesn't matter, but for maths,
when its assumptions are pressed, is incomplete with any foundation.

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