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Author Topic:   Is mathematics a science?
nwr
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Posts: 6408
From: Geneva, Illinois
Joined: 08-08-2005
Member Rating: 5.1


Message 1 of 48 (238030)
08-28-2005 4:24 PM


This is being much discussed in Which came first: the young earth, or the inerrant scripture?, where it is way off topic. I suggest a new thread, perhaps in Is It Science? for continuing the discussion of mathematics.
I will take the position that mathematics is not a science, since the word "science" has come to mean an empirical study.
The role of mathematics within science can be explored in this thread. That would allow addressing the question of whether mathematics is a language, as suggested in Message 108.

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AdminNosy
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Message 2 of 48 (238393)
08-29-2005 8:53 PM


Thread moved here from the Proposed New Topics forum.

  
Maxwell's Demon
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Posts: 59
From: Stockholm, Sweden
Joined: 05-09-2004


Message 3 of 48 (238442)
08-29-2005 10:33 PM
Reply to: Message 1 by nwr
08-28-2005 4:24 PM


Well...
In science we start with observations about the natural world.
We construct a theory by means of induction (generalising from a few basic principles).
If true, the theory aught to tell us more about the world than the observations that spawned it (or it wouldn't be a generalisation). These are predictions, and can be tested for. By doing so we see if our deduced theory was sound.
In math we start with axioms about an abstract world (where things like perfect circles exist).
We construct a proved theorem by means of deduction (the conclusion follows necessarily from the stated premises).
The proved theorem is true.
I'd say science works by means of induction, and math by means of deduction.
Science tells us things about the natural world, and mathematics about an abstract world where certain axioms hold true.
This message has been edited by Maxwell's Demon, 08-29-2005 10:35 PM

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Nuggin
Member (Idle past 2492 days)
Posts: 2965
From: Los Angeles, CA USA
Joined: 08-09-2005


Message 4 of 48 (238450)
08-29-2005 11:00 PM
Reply to: Message 3 by Maxwell's Demon
08-29-2005 10:33 PM


Math as a language
I agree.
I would suggest, as I did in the other thread, that Math is more like a language than a science.
It's rigidly structured. It has it's own characters. It can be used to describe other things. You can exchange ideas using math as your only means of communication.

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nwr
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Posts: 6408
From: Geneva, Illinois
Joined: 08-08-2005
Member Rating: 5.1


Message 5 of 48 (238455)
08-29-2005 11:25 PM
Reply to: Message 3 by Maxwell's Demon
08-29-2005 10:33 PM


I would say you are about right there.
There are times when mathematicians are thinking a bit like scientists, doing experimentation (with mathematical objects). So it isn't all deduction from axioms. However, what is observed from experimentation is never considered sufficient to constitute a proof. For that one needs deduction.

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nwr
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Posts: 6408
From: Geneva, Illinois
Joined: 08-08-2005
Member Rating: 5.1


Message 6 of 48 (238457)
08-29-2005 11:27 PM
Reply to: Message 4 by Nuggin
08-29-2005 11:00 PM


Re: Math as a language
Several people have said that math is a language. But I think mathematicians might disagree with that.
Of course, mathematics includes its own notational language. But mathematics is very much about methods and procedures, and some of those methods do prove useful in science.

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Dr Jack
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From: Immigrant in the land of Deutsch
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Message 7 of 48 (238509)
08-30-2005 6:01 AM
Reply to: Message 4 by Nuggin
08-29-2005 11:00 PM


Re: Math as a language
I would suggest, as I did in the other thread, that Math is more like a language than a science.
Maths is not a language. Language has no problem solving capability; maths does.
It's rigidly structured. It has it's own characters. It can be used to describe other things.
Granted.
You can exchange ideas using math as your only means of communication.
Unless those ideas are specifically about maths; no, you can't and even then you will struggle.

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Dr Jack
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Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 8 of 48 (238510)
08-30-2005 6:03 AM
Reply to: Message 3 by Maxwell's Demon
08-29-2005 10:33 PM


You know, I wonder whether maths has changed from being a science to something else. While the modern mathematically synthesis is derived from axiomatic routes; it has it route fairly and squarely in an emprical description of the world.
Consider the row over negative and imaginary numbers (but what do they mean?) and even zero in long distant times.

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Chiroptera
Inactive Member


Message 9 of 48 (238541)
08-30-2005 8:46 AM
Reply to: Message 1 by nwr
08-28-2005 4:24 PM


definition of science
From Merriam-Webster:
1 : the state of knowing : knowledge as distinguished from ignorance or misunderstanding
2 a : a department of systematized knowledge as an object of study b : something (as a sport or technique) that may be studied or learned like systematized knowledge
3 a : knowledge or a system of knowledge covering general truths or the operation of general laws especially as obtained and tested through scientific method b : such knowledge or such a system of knowledge concerned with the physical world and its phenomena : NATURAL SCIENCE
4 : a system or method reconciling practical ends with scientific laws
5 capitalized : CHRISTIAN SCIENCE
Under definition 2 (especially 2a) mathematics can be described as a science, definitely.
However, when I think of "science" I am thinking of definition 3; in this case, I would say that mathematics is not a science.

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Brad McFall
Member (Idle past 5032 days)
Posts: 3428
From: Ithaca,NY, USA
Joined: 12-20-2001


Message 10 of 48 (239089)
08-31-2005 3:47 PM
Reply to: Message 1 by nwr
08-28-2005 4:24 PM


let discovery begin!
quote:
Appendix B
Ars Combinatoria
(some German)
1. Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind — as Eddington put it — the colorful tale of the subjective storyteller mind. In Appendix A we have discussed the structure of mathematics as such. The present appendix deals with some of thesimplest structures imaginable, the combinatorics of aggregates and complexes. It is gratifying that this primitive piece of symbolic mathematics is so closely related to philosophically important problems of individuation and probability, and that it accounts for some of the most fundamental phenomena in inorganic and organic nature. The same structural viewpoint will govern our account of the foundations of quantum mechanics in Appendix C. In a widely different field J. von Neumann’s and O. Morgenstern’s recent attempt to found economics on a theory of games is characteristic of this trend. The network of nerves joining the brain with the sense organs is a subject that by is very nature invites combinatorial investigation. Modern computing machines translate our insight into the combinatorial structure of mathematics into practice by mechanical and electronic devices.
It is in view of this general situation that we are now going to insert a few auxiliary combinatorial considerations of an elementary nature concerning aggregates of individuals. The reader should be warned beforehand that in their application to genetics the lines are drawn somewhat more sharply than the circumstances warrant. In the progress of science such elementary structures as roughly correspond to obvious facts are often later recognized as founded on structures of a deeper level, and in this reduction the limits of their validity are revealed. This hierarchy of structures will be illustrated in Appendix D by the theory of chemical valence.
page 237-8 in PHILOSOPHY OF MATHEMATICS AND NATURAL SCIENCE by Hermann Weyl Princeton Uni Press 1949
If one thinks one can understand the reduction of said sharp genetic line(s) in terms of a hierarchics of valences (contingent nonetheless) then the math that bridges this text of Weyl could be considered science. I often think I am permanently in such possession. But that is such (relative to this thread head) IF the computer science structure Weyl passed on can be empirically such as to reduce the same structure. I do not think that it does. Hence math is not science. A philosophical position from Wolfram’s new kind of science however might have one think (thought previously) that it is (so irreducibly complex) but again philosophy is not math or science now is it? Theology is not teleology.
quote:
By breeding experimens one has succeeded in dissolving the genetic consitution in to an aggregate of individual genes or 'points', much like as chemistry dissolves a molecule into an aggregate of atoms
op.cit. p240
Viscosity was not the word but the math could cover it.
Empirical mathematics and breeding empirical mathematicians are two different combinations. Experimental space both contain.
There is a metaphor mathematically where Weyl used a simile.

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crashfrog
Member (Idle past 1467 days)
Posts: 19762
From: Silver Spring, MD
Joined: 03-20-2003


Message 11 of 48 (239156)
08-31-2005 5:42 PM
Reply to: Message 7 by Dr Jack
08-30-2005 6:01 AM


Re: Math as a language
Maths is not a language. Language has no problem solving capability; maths does.
Math is simply a symbolic means by which statements comprised of symbols are either accepted or rejected, based on whether or not they can be constructed through a process of certain valid transformations from certain a priori axioms.
So it is a language; that is, a symbolic representation of information with grammar. Math has a very specific, rigid grammer, in fact. The grammar of math can be studied and operated with no recourse to the actual referents described by the symbols.
For instance, 2 + x = 4 can be solved for x without recourse to the actual meaning of any of those symbols; that's what makes computation possible. Your computer does math not because it knows what "2" or "+" mean but because it's been programmed with grammatical rules that tell it what transformations to apply to that string of symbols.
Unless those ideas are specifically about maths; no, you can't and even then you will struggle.
Not so. The very fact that you're sitting at a computer that does math proves that this isn't so, and in fact, is a pretty good indication that math is something different than science.
But to me it's pretty cut-and-dry. Mathematical knowledge is supported deductively; scientific knowledge is supported inductively. Two different worlds.
Math is not science.

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nwr
Member
Posts: 6408
From: Geneva, Illinois
Joined: 08-08-2005
Member Rating: 5.1


Message 12 of 48 (239319)
09-01-2005 12:47 AM
Reply to: Message 11 by crashfrog
08-31-2005 5:42 PM


Re: Math as a language
crashfrog writes:
Math is simply a symbolic means by which statements comprised of symbols are either accepted or rejected, based on whether or not they can be constructed through a process of certain valid transformations from certain a priori axioms.
There is a lot more to mathematics than symbol manipulation. Many of the most important mathematical research papers consist mainly of informal prose with relatively little use of symbols.
The very fact that you're sitting at a computer that does math proves that this isn't so, and in fact, is a pretty good indication that math is something different than science.
Roger Penrose wrote two books "The Emperor's New Mind" (1989) and "Shadows of the Mind" (1993), in which he claims to disprove the possibility of AI. The basis of his argument is that computers cannot do math.
The general view is that Penrose's arguments don't work. The critics point to flaws in his claimed proof. However, I think it fair to say that nobody has proved that computers can do math.
My point -- there is a large difference between doing computation, and doing mathematics. Your claim that math is a language is far from proven.

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Dr Jack
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Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 13 of 48 (239398)
09-01-2005 6:02 AM
Reply to: Message 11 by crashfrog
08-31-2005 5:42 PM


Re: Math as a language
Computers do not operate by doing maths. Believe me, I know - I have a masters in mathematics and am a computer programmer by trade; I'm intimately familiar with both.
What computers acheive is done through boolean logical operations, not mathematics. The meaning you perceive is there, not because it is communicated by mathematics but because it is constructed as a visual image that you understand; whether through visual or textual means. And is these that communicate the meaning not the underlying logic.

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Brad McFall
Member (Idle past 5032 days)
Posts: 3428
From: Ithaca,NY, USA
Joined: 12-20-2001


Message 14 of 48 (239406)
09-01-2005 7:03 AM
Reply to: Message 13 by Dr Jack
09-01-2005 6:02 AM


Re: Math as a language
Nice.
Jack,
I could not have said it better myself!

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Chiroptera
Inactive Member


Message 15 of 48 (239476)
09-01-2005 10:33 AM
Reply to: Message 12 by nwr
09-01-2005 12:47 AM


Whitehead and Russell
quote:
Many of the most important mathematical research papers consist mainly of informal prose with relatively little use of symbols.
This is true, because mathematicians think in terms of the concepts (and, like anyone else, would find reading a paper full of nothing but abstract symbols rather dry). However, in principle, any paper in pure mathematics can be translated into pure symbolic logic, with symbols representing the various objects and relations, and each step being due to a specific rule of logic.
At the beginning of the last century, Russell and Whitehead began a program to translate all of mathematics into pure symbolic logic; it turned out to be much more involved than they had anticipated and couldn't come close to translating all of mathematics. But they did show that ultimately, all mathematics is subfield of logic.
-
quote:
There is a lot more to mathematics than symbol manipulation.
The prose that you see in a paper in pure mathematics is describing what is actually symbol manipulation. Nonetheless, you are correct; it does take a certain amount of creativity and intuition to figure out what is an important enough theorem to prove, and to figure out the path (the exact manipulations of symbols, or the prose description of those manipulations) the proof is going to take.

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