Science And Math Was: Python's Lisp heritage

[Boris Borcic]

Tim Daneliuk wrote

> 
> I disagree strongly (but am willing to be convinced otherwise).  All of
> mathematics is a formal construct of the human mind created with the
> intent of removing ambiguity and enhancing our ability to describe what
> we *think* and what we *observe*.


(1) one may well deny you the adequacy of speaking of "the human mind"
in that manner, esp. if you further deny its existence as a bona-fide 
component of nature. Same remark for your use of "we".

(2) the nature/position/interpretation of ambiguity would deserve 
volumes by itself. Most people's understanding of the distribution of 
ambiguity is biassed by having it confused with cases of visible impact.

(3) I believe it's you who use in another post of this thread, the 
classical question "Is mathematics discovered or invented ? It's clearly 
invented". To me, such rhetorics display some lack of sophistication as 
regards the ambiguity of the value of single words/concepts.

My reply to it : consider *misunderstandings* as objects; clearly they 
have scientific status, and simple misunderstandings recur 
isomorphically. Clearly they are creations of "the human mind", as they 
don't exist and can't be described without rooting parts of any 
description of a misunderstanding to human minds. But are they 
"discovered" or "invented" ? Neither word applies; this kind of 
disproves your implied notion about innovation, that it should be either 
"discovered" or "invented".

My belief is that all this debate is very much complicated by "bootstrap 
notions" about the nature of understanding and knowledge, that need to 
be sorted out before any serious discussion is possible. Mathematics is 
indeed very much concerned with building the means to describe solutions 
to misunderstandings (if not the misunderstandings themselves). That's 
what imho you allude to when referring to "ambiguity", but this way of 
putting it kind of implies the confusion of misunderstandings with 
"misunderstandings traceable to the effects of spot lexical 
ambiguities". My understanding of Godel's theorem is precisely that 
there's much more to ambiguities than spot ambiguities of a single 
lexeme unit or word.

Nothing forbids to describe natural science in exactly the same terms, 
to say that it is concerned with destroying ambiguities, and solving 
eventual misunderstandings : calling repeatable experiments in as 
witnesses. Is being witness to a classical experiment of physics that 
much different from being witness to a proof of some classical math 
theorem ? Not to me.

> We can (and do) create all kinds of new
> calculi that are problem- or domain-specific.  That is, we *invent*
> mathematics to suit our needs - mathematics is not one-for-one
> correspondent to the natural universe.


To me what you are really saying is that mathematics can't claim to have 
a special adequacy to the natural universe, any different in essence 
than natural language. I mean that's how I need to translate your 
opinion to make it more or less my own.


> The reason I take this position is pretty simple.  We are able to construct
> mathematical systems which describe intellectual abstractions for which
> there is no natural analog.


This line of reasoning is imho biassed by a closed-world assumption that 
doesn't apply. Your sentence should really end with "...no natural 
analog I can spot as such".

> That is, mathematics can embrace way more than just the natural world

> and its workings.


I'd rather say that mathematics supplements our natural senses to the 
point of allowing us to touch matters that our (other) natural senses 
and languages do not allow us to touch or share (for various reasons).


> Similarly, a mathematician
> can introduce ideas like an "Incompleteness Theorem" which is a meta-mathematical
> commentary on mathematics itself - something far beyond the domain and range of
> mere natural science.


As I've already alluded above, my belief is that the Goedel Theorems 
have an interpretation in non-mathematical terms, as the idea that 
trying to kill lexical ambiguity by mechanically hardening the 
definitory relationship between lexemes is doomed to fail (in some 
situations) because the whole system of lexemes then becomes ambiguous, 
allowing coordinated reinterpretation of many lexemes in such a manner 
that the mechanical definitions stay invariant.


> 
> I'm not choosing sides here on the math vs. science debate - both are important
> artifacts of the human intellect.  I merely take umbrage with the notion that
> mathematics is innately wired into the universe somehow.  


I see what you mean, and if I am to choose corresponding umbrage with 
something that seems to base your notion, it's the idea that mankind 
could be ascribed the role of a single cognizer that can be contrasted 
to the universe as a whole.

Regards, Boris Borcic
--
"God is Truth too complicated for children, and that Mankind failed
  to teach itself ever since"