[FOM] Mathematics ***is*** formalising of our thought and intuition

[Robert Lindauer]

a) I thought you were claiming mathematics was -the only science-
creating formal tools for making our thought and intuition powerful.
  As I said, many other kinds of scientists contribute to the effort
in different ways, your definition turns their work into mathematics,
which is simply not true.    Political theory also creates formal
models for political situations and is not -ipso facto- mathematics.
Political, Physical and other scientists use mathematics where
appropriate and use other formalizations where mathematics is less
appropriate (e.g. Linguistics has formal models of phonetical systems,
calling them "mathematics" because available for formal modeling is at
best a stretch - like saying that game programming -is mathematics-).
Meta-mathematics (both formal and informal) is not necessarily
mathematics and yet does definitely create and use formal methods of
consideration (e.g. Principia Mathematica).

b) My claim was that mathematics was not -necessarily- formal and I
think that's borne out by the history of mathematics and even the
dominant practice of mathematics in this century (e.g. by
non-university mathematicians like my 10-year-old-son) - that there is
a localization of formalization in Western Civilization's university
system since the turn of the previous century does not overturn
thousands of years of informal or quasi-formal mathematics!  What is
necessary is the method of abstraction by which we consider things
without regard to their particular character  - that is, without
regard to -what it is we're talking about at all- - this is 'almost
formalization' in that we are considering things "formally" in the
sense of not considering them concretely but not "formally" in the
sense of defining method for consideration.  Other sciences do not
completely abstract the subject-matter (neither does mathematics
'always'...).

d) regarding the "guarantee" - even finite and feasible-sized proofs
are subject to the size problem.  "The largest compact feasible proof
is not verifiable" where compact means that any -other proof- must be
larger than the original proof.  But this misses the point which you
effectively make for me, much of mathematics is not feasible, making
it unreliable, you are in the minority, as I understand it, about your
wanting only feasible proofs about feasible mathematical objects.
(However much I may agree with you.)  In any case, those people who
disagree with you are -also mathematicians doing mathematics-.

e) the need for formal presentation in other sciences is clear -
without formalization, a presentation is incomplete, unconvincing and
unclear and simply stands in need of a formal presentation.  This
bears on the formal character of mathematics too - Cantor's system's
formalization (zfc-ish) is not much different from the formalization
of phonetics in that the original theories were formalizable but not
formalized and progress in the sciences leads to this kind of
clarification after the initial work is done.  In mathematics this
doesn't mean that the initial work isn't mathematics (it is!), and in
other sciences that doesn't mean that the formalizing work is
mathematics (it isn't -always-!)

f) the formal aspects of rhetoric are as interesting as the formal
aspects of logic, e.g. "Uses of Argument" by Toulmin.

Best,

Robbie Lindauer



On Sat, May 29, 2010 at 1:48 PM, Vladimir Sazonov
<vladimir.sazonov at yahoo.com> wrote:
>>From: Robert Lindauer rlindauer at gmail.com
>> On May 28, 2010, at 3:20 PM, Vladimir Sazonov wrote:
>>
>>
>> VS:
>> Mathematics is indeed ***the only*** science or a kind of engineering
>> creating formal tools for making our thought and intuition powerful;
>> [see also a comment on Computer Science]
>>
>> RL:
>> If the other sciences weren't -there- then mathematics would be useless and pointless.
>
>
> I do not understand your conclusion. Formal tools strengthening
> our thought (on some imaginary mathematical worlds) are potentially
> universally applicable just because of their formal nature as I
> argued in my last posting.
>
>
>> In any case, any  "relatively un-biased" (for instance,
>> not only by pure mathematicians) examination of a standard
>> university will show that there are lots of other sciences.
>
>
> I absolutely do not understand how my description of math
> contradicts to existence of other sciences.
>
>
> Mathematics makes our thought and intuition powerful not
> only by training (as any other science also do) but mainly
> by creating formal tools. It is similar to creating a
> bicycle which makes us faster or nuclear power station
> which makes us more powerful. One example of such a formal
> (in a wide sense of this word) tool created in mathematics
> is the Calculus. Other sciences can use these tools, but
> only for mathematics creating these formal tools of
> thought is the main goal. Of course, sometimes physicists
> participate in creating such formal tools with the most
> great example of Newton. But at that moment Newton behaved
> as an great mathematician.
>
>
>> VS:
>> as I wrote in a recent
>> posting where, also as Hendrik suggests, "long sequences of reasoning
>> steps can be carried out without error" with a guarantee.
>>
>>
>> RL:
>> There is no such guarantee in mathematics.  Consider
>> the "long problem" problem (from FOM a few years ago):
>>
>>
>> Let's imagine that there is a proposed proof of a theorem T,  P.
>> Let's imagine that P is 10^10^10^10 steps long.
>> Let's imagine that there is a proof that no proof of the theorem could be shorter than P's length.
>
>
>
> Let me recall that I wrote (on Tue May 25 17:43:54 EDT 2010)
>
>
>
>      any (formal) mathematical proof must be of a feasible length
>
>
>
> Otherwise this is an imaginary proof. You cannot present to a mathematical
> journal an imaginary paper with imaginary proofs. Only meta-mathematics can
> deal with such imaginary proofs like mathematics can deal with huge numbers
> and with infinite objects and other as imaginary things. But when we prove
> something meta-mathematical this proof must be again feasible.
>
>
>
> Mathematicians usually check formal correctness of their proofs
> without invoking computer since it is just an intellectual endeavour.
>
>
>
> Let me stress again that I understand "formal" in a wide sense
> of this word and sometimes in the narrow. When I want to emphasise
> the latter I use "potentially formalisable" because "absolutely
> formal" is unnecessary and often useless for the usual mathematics,
> although "potential formalisability" is a fair characterisation of
> the contemporary level of mathematical rigour.
>
>
> However, I use "potential" not in the ordinary sense of "potential infinity"
> which would allow considering imaginary finite but not feasibly existing
> formal derivations. "Potentially formalisable" means that the full
> formalisation could be possibly very long, but still feasible.
>
>
> I believe that this usage of terms is reasonable and not over-complicated.
> I also believe that this resolves your concerns on correctness guarantee
> and computer verification of potentially formalisable proofs. Everything
> remains in mathematics as usual. I only present a formalist view.
>
>
> In this view I am not very interested in Hilbert program of proving
> consistency (by weak finitistic methods). I am interested in an
> appropriate reasonable formulation of formalist view as such.
>
>
>
>>
>> VS:
>> In all other
>> science, even in Physics (and least of all in Political Sciences and
>> Philosophy), long (and even not so long) sequences or reasoning steps
>> are much less reliable.
>>
>>
>> RL:
>> A formal proof in Physics or chemistry is just as accurate
>> as a formal proof in mathematics but uses additional assumptions
>> (molecular bonding, etc.),
>
>
> A physicist will ignore any formal mathematical tool in favour
> of physical truth. Their goal is truth in the real world and
> not the level of rigour how it was obtained. They are happy to
> use rigour (i.e. formal mathematical tools) but only as much as
> it helps to get truth --- a normal approach of all people to
> the instruments.
>
>
> On the other hand, the goal of mathematics (from the formalist
> point of view as I describe it) is not a truth at all. The goal
> is formalising our thought and intuition, i.e. creating
> formal tools of thought both for other sciences and for using
> in mathematics itself (to create other formal tools on the base
> of already created).
>
>
>
> the additional need for empirical
>> verification actually gives something better than what a purely
>> formal proof can give, namely soundness.
>
>
>
> If a formal tool of thought created in math is really a *tool of thought*
> then it is evidently sound in the sense that it is indeed helpful to
> some our specific style of thought. (The mechanism how it works is
> inevitably hidden here since we discuss on a very general level.)
> If this tool and corresponding thought domain or thought style is
> applicable to a real situation in an empirical science then it will
> demonstrate it soundness in this concrete situation just by the very
> fact of applicability. But in any case, the point is that the tool
> is formal and when we check correctness of a some parts of this tool
> or of some "runs" of this tool (such as formal derivation) we only
> should check formal correctness independently on any truths.
> Since applicability to any empirical domain is unknown in advance
> and even can be permanently doubted even after a lot of experience,
> those who apply should permanently check on the suitability of this
> tool and be always ready to change this tool or even to be prepared
> to suggest any ideas on the desirable new tool.
>
>
>
>>
>> Surely mathematics
>> is foundational in that every science must make some use of
>> mathematical concepts and procedures,
>
>
>
>
> Why must? Formal tools suggested by mathematics ***can*** be used
> if they are suitable.
>
>
>
> however, it does not follow that the
>> whole content of those other sciences (e.g. chemistry) is
>> mathematical!
>
>
>
> Who said that?
>
>
>>
>>
>> VS:
>> Here I completely agree. No other science creates formal tools
>> for thought.
>>
>>
>> RL:
>> Logic obviously does, metalogic comes to mind, rhetoric as well.
>>
>
>
>
> I would  only refer here to mathematical logic which is a part
> of mathematics. Which formal tools does rhetoric create (on a
> permanent basis, as a way of its existence and not as some episodic
> occasion)?
>
>
>
>>
>>
>> Or, think more plainly, pure mathematics is the systematic
>> consideration of proto-mathematical concepts (space/time, infinite,
>> finitude, recursion, grouping, shape, etc.) without regard to the
>> conceptual content (that is, without regard to the concrete concept
>> itself or the things to which they are to be applied).
>
>
>
> Mathematical rigour ( = formal character) is so unique and central
> to mathematics that I wander how it is possible to omit mentioning
> it at all.
>
>
>
>>
>>
>> RL:
>> The "formal" nature of mathematical reasoning is characteristic
>> (but not definitional) of reasoning about abstractions of things
>> (viz things without respect to their individual qualities),
>> however, it is neither necessary nor sufficient for mathematical science.
>> Consider  Cantor's Grundlagen ("mathematics" but not "formal"),
>> or Wittgenstein's Tractatus (not mathematics, but "formal").
>
>
> I do not know about Wittgenstein, but Cantor's set theory was
> eventually formalised. Calculus (Analysis) was also existing
> quite a long time weakly-formal till epsilon-delta approach,
> Dedekind cuts and later as the Non-standard Analysis. The general
> thesis is that any mathematical consideration is potentially
> formalisable. This is the goal of mathematics confirmed by its history.
>
>
>>
>>
>> Obviously "formal" is a characteristic
>
>
>
> Something optional? Please show any contemporary mathematical
> proof which is inherently non-formalisable but was accepted to a
> mathematical journal and thereby was not considered by
> mathematical community as highly suspicious or just as a mistake.
>
>
>
> of considering things
>> in complete abstraction
>
>
>
> What means in "complete abstraction" if it is not considering formally?
>
>
>
> (without respect to the individual
>> qualities of the things considered) - HOWEVER - what Hegel's
>> definition captures that this definition fails to capture
>> is the essential ground of proto-mathematical pre-formal
>> concepts in to considerations of abstractions of those
>> pre-formal concepts in a -generally scientific- or systematic way.
>>
>>
>> Without the proto-mathematical ideas of, for instance,
>> grouping, space, time, motion, iteration, demonstration, etc.
>> we would have none of set theory, topology, calculus,
>> geometry, algebra, or proof theory, etc.,  because without
>> these foundational pre or proto-mathematical notions,
>> the mathematics describing them is simultaneously meaningless and useless.
>
>
>
> Creating formal tools for thought (if they are REALLY tools helping
> our thought assuming that this thought is about something presenting
> interest) cannot be meaningless and useless. It is not necessary to
> mention all these particular proto-mathematical ideas because the
> style of thought which is capable to be formalised will most probably
> deal with these or some others, which we cannot know in advance,
> proto-mathematical ideas. Any explicit listing in a general
> definition is restrictive and ad hoc.
>
>
>
> Also abstraction is not belonging exclusively to mathematics.
> Thus it is not a definitive feature, unlike mathematical rigour
> --- the innate formal nature --- which is pursued so eagerly
> and with so passion by no other science (even if it is regularly
> using mathematics).
>
>
> Best wishes,
>
>
> Vladimir
>
>
>
>
>



-- 
It gives.