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Author | Topic: Is mathematics a science? | |||||||||||||||||||||||||||
nwr Member Posts: 6409 From: Geneva, Illinois Joined: Member Rating: 5.3 |
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 Administrator Posts: 4754 From: Vancouver, BC, Canada Joined: |
Thread moved here from the Proposed New Topics forum.
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Maxwell's Demon Member (Idle past 6249 days) Posts: 59 From: Stockholm, Sweden Joined: |
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 2513 days) Posts: 2965 From: Los Angeles, CA USA Joined: |
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 Member Posts: 6409 From: Geneva, Illinois Joined: Member Rating: 5.3 |
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 Member Posts: 6409 From: Geneva, Illinois Joined: Member Rating: 5.3 |
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 Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.3 |
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 Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.3 |
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 |
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 5053 days) Posts: 3428 From: Ithaca,NY, USA Joined: |
quote: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: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 1487 days) Posts: 19762 From: Silver Spring, MD Joined: |
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: 6409 From: Geneva, Illinois Joined: Member Rating: 5.3 |
crashfrog writes:
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.
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. 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 Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.3 |
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 5053 days) Posts: 3428 From: Ithaca,NY, USA Joined: |
Nice.
Jack, I could not have said it better myself!
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Chiroptera Inactive Member |
quote: 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: 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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