Re: Epistemology 201: The Science of Science

From: Jason (jasonstevensNOSPAM_at_free.net.nz)
Date: 02/06/05 

Date: Sun, 06 Feb 2005 14:13:34 +1300

Lester Zick wrote:

> On Sat, 05 Feb 2005 10:54:14 +1300, Jason
> <jasonstevensNOSPAM@free.net.nz> in comp.ai.philosophy wrote:
>
>
>>[snip]
>>
>>
>>>>for rational and irrational concepts. No point to
>>>>defining transcendental as irrational.
>>>
>>>
>>>It's not *defined* as irrational, it follows from the definition of
>>>"transcendental" that such numbers also meet the requirements of
>>>"irrational".
>>
>>My 2 cents:
>
>
> Always welcome.

Thanks.

>
>
>>There are many more numbers than names, so there will always be nameless
>>numbers no matter how clever the notation is. With algebraic notation,
>>the "transcendentals" transcend our ability to finitely describe them
>>with expressions of roots. All we can do is give them names, like 'pi'
>>and 'e'.
>
>
> But we can describe them algebraically even if we can't finitely
> describe their roots. So use of the term algebraic is ambiguous.

You're right, I think they term the numbers 'transcendental' beyond the
point where they can be expressed by finite algebraic roots - ignoring
the infinte sums and the like.

>>Hence all transcendentals are irrational, because all rational numbers
>>can be described algebraically. So yeah, they aren't *defined* as
>>irrational, like you say.
>
>
> Not best evidence because we have a better criterion. All rationals
> can be integrally related which is why they're called rationals and
> they can be pointed out on straight lines with right angles for that
> reason. There are also things which can be pointed out on straight
> lines with right angles which are not rational and are called
> irrational in consequence.
>
> Then there numbers which cannot be pointed out on straight lines
> using right angles which are called transcendental as a result. The
> inability to finitely describe the roots of transcendentals in algebra
> just reflects the inability to point them out on straight lines using
> right angles.
>
> This resolves the tautological riddle for rational, irrational, and
> transcendental numbers and allows us to integrate geometric and
> algebraic concepts.

I like the approach from the limits of language. Language and rationals
('ratios' a/b) are countable sets, but there are too many irrationals to
count. The invention of the root captured a few, but the ones that got
away are the transcendentals. If they can be named, then strictly
speaking they are no longer transcendental, but maths has fixed the term
it seems.

>>If you throw a dart at the number line, you will hit a transcendental
>>number with probability 1. Well, (1 - infinitesimal) or (1 - 1/infinity).
>
>
> My two cents -
>
> If you throw a dart at a straight line you will hit an irrational or
> rational number. You have no chance of hitting a transcendental
> number because they aren't there.

Well, they are only there if we invent a language to try and describe
numbers. But then... if the universe turns out to be descrete, then
you'll be right.

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... All rationals ... There are also things which can be pointed out on straight ... > inability to finitely describe the roots of transcendentals in algebra ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... All rationals ... There are also things which can be pointed out on straight ... > inability to finitely describe the roots of transcendentals in algebra ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... >with expressions of roots. ... All rationals ... There are also things which can be pointed out on straight ... inability to finitely describe the roots of transcendentals in algebra ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... >with expressions of roots. ... All rationals ... There are also things which can be pointed out on straight ... inability to finitely describe the roots of transcendentals in algebra ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... >with expressions of roots. ... All rationals ... There are also things which can be pointed out on straight ... inability to finitely describe the roots of transcendentals in algebra ...
    (sci.math)