Is maths a science?

[Derleth]

@David Simmons

I’m not really qualified either. On these boards I once wrote that math was a way for scientists, and others, to deduce guaranteed results from a set of postulates. Several mathematicians immediately pointed out that on the frontiers of mathematics things are not as certain as I thought.

Hm. I’m not quite sure what they would have been talking about. I’m familiar with Gödel’s incompleteness theorems, but to my knowledge those don’t detract from the ability of mathematics to make formally, rigorously verifiable statements within the framework of a self-consistent axiomatic system.

I’m probably decades out of date here. Gödel did this work in the 1930s, and I’m sure more recent mathematics has lead to more interesting results. I’m curious to find out, and I hope one of our resident mathematicians will enlighten us.

[scm1001]

@Jeff Lichtman

Math is deductive, while science is inductive. That is, math builds knowledge by deducing theorems using the rules of logic, while science builds knowledge by inducing theories from observation. .

there seem to be many mathematical theorems that come closer to induction than deduction. Eg. both Fermats last theorem and the four colur map problem were for hundreds of years only known to be possibly true because of induction, e.g. tried many times, couldn’t find a counterexamples. I am not a mathematician, so I dont know if for instance the four colour thereom was solved by deduction or by merely trying all the possibilities (induction)

[Thudlow_Boink]

Mathematicians often use inductive reasoning to investigate what’s probably true, but (unlike science—or should I say unlike other sciences?) then you have to actually prove it using deductive reasoning before it’s accepted as indisputably true.

Part of a deductive proof may involve considering all cases or “trying all the possibilities.”

Warning: Mathematicians sometimes use a proof technique known as “mathematical induction.” This is not the same as the inductive reasoning we’ve been talking about in this thread. A proof based on mathematical induction certainly can be valid and logically airtight.

[Colibri]

@Gary T

I’m seeing a lot of excuses to explain why math isn’t a science, and I don’t find any of them persuasive. At its essence, science is organized knowledge. Look at a dictionary under “science” and see if math doesn’t fit most of the definitions. Now look under “mathematics” – my dictionary starts the definition off with “the science of numbers…”

If you want to define science such that math doesn’t qualify, help yourself. But what come across as blanket statements saying in effect “math cannot be a science because…” seem to show limited understanding of both terms.

Not really. There are simply different definitions of “science” involved here. Mathematics is a science in the sense of an organized body of knowledge. In the stricter sense of a field of knowledge that is subject to empirical verification (which I was careful to spell out in my first response to this thread), it is not a true science. I don’t think that most professional scientists (in the strict sense) would recognize mathematics as a science. I am not sure whether professional mathematicians regard their field as a science, or as something separate.

[Humanist]

@scm1001

there seem to be many mathematical theorems that come closer to induction than deduction. Eg. both Fermats last theorem and the four colur map problem were for hundreds of years only known to be possibly true because of induction, e.g. tried many times, couldn’t find a counterexamples. I am not a mathematician, so I dont know if for instance the four colour thereom was solved by deduction or by merely trying all the possibilities (induction)

The four color theorem was proven with the aid of a specially written computer program. They logically divided all types of maps that would prove the four color theoroem into something like 60 types of maps, and used a computer to show that none of these types could logically exist.

Fermat’s last theoroem has been proven by deduction, using (to my recollection) semistable curves. It’s unlikely that this was the method Fermat used (since they are a recent extrapolation), but it has been proven.

[FriendRob]

@David Simmons

I’m not really qualified either. On these boards I once wrote that math was a way for scientists, and others, to deduce guaranteed results from a set of postulates. Several mathematicians immediately pointed out that on the frontiers of mathematics things are not as certain as I thought.

I’d argue that math is not only a science, it’s an experimental science.

Think about geometry. The postulates are statements about points, straight lines, etc. But “point”, “line”, etc. are never defined. We have to rely on an intuitive notion of these items in order to proceed. We can only know if our intuitive notions and postulates are correct by applying them and checking if they agree with the “real world”.

@ultrafilter

Math(s) is an art, not a science.

Science is an art, not a science.

[ultrafilter]

@FriendRob

I’d argue that math is not only a science, it’s an experimental science.

Think about geometry. The postulates are statements about points, straight lines, etc. But “point”, “line”, etc. are never defined. We have to rely on an intuitive notion of these items in order to proceed. We can only know if our intuitive notions and postulates are correct by applying them and checking if they agree with the “real world”.

Geometry doesn’t care what a point, line, or plane is. All it cares about is the properties those things have.

The verification that geometry matches up with the real world is outside the field of geometry, and math in general.

[Thudlow_Boink]

@FriendRob

Think about geometry. The postulates are statements about points, straight lines, etc. But “point”, “line”, etc. are never defined. We have to rely on an intuitive notion of these items in order to proceed. We can only know if our intuitive notions and postulates are correct by applying them and checking if they agree with the “real world”.

Actually, that’s kinda backwards from the way a pure mathematician would think of it. “Point” and “line” don’t have to mean anything or correspond to anything in the “real world.” It may help us to think about them if we can visualize them, but we don’t have to in order to prove things about them. We can start with whatever axioms we like, whether or not they correspond to anything in the real world (and knowing whether or not they do is the task of science, not of math).

@Albert Einstein

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

@Bertrand Russell

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

[nivlac]

As already pointed out, whether math is a science will depend on your definition of “science”. So I’m not going there. I also agree that most professional mathematicians do not view themselves as scientists. However, from a practical standpoint, I would say that most lay people would view math as a science. I work with many mathematicians within my business (outside academia) – their titles? Scientists.

[Chronos]

Quoth Humanist:

The four color theorem was proven with the aid of a specially written computer program. They logically divided all types of maps that would prove the four color theoroem into something like 60 types of maps, and used a computer to show that none of these types could logically exist.

Fermat’s last theoroem has been proven by deduction, using (to my recollection) semistable curves. It’s unlikely that this was the method Fermat used (since they are a recent extrapolation), but it has been proven.

I don’t think that’s what scm1001 meant. Now, both of those statements have been deductively proven true, but there was a time before that when they were believed true but not proven. Or, to use a contemporary example, the Goldbach conjecture. It’s been tested for numbers up into the billions, and for those numbers, has always turned out to be true. Because of this, most mathemeticians inductively conclude that it’s probably true for all n. But this has never been deductively proven. To a scientist’s point of view, the Goldbach conjecture has been proven, at least to the extent that anything in science can be proven, but the “proof” is entirely inductive.

[David_Simmons]

@FriendRob

I’d argue that math is not only a science, it’s an experimental science.

Think about geometry. The postulates are statements about points, straight lines, etc. But “point”, “line”, etc. are never defined. We have to rely on an intuitive notion of these items in order to proceed. We can only know if our intuitive notions and postulates are correct by applying them and checking if they agree with the “real world”.

From the point of view of mathematics does it really matter whether or not its operations agree with the real world? That is of great importance to physical scientists, engineers and other technictians but from a strictly mathematical viewpoint who cares?

[Colibri]

@FriendRob

I’d argue that math is not only a science, it’s an experimental science.

Kindly explain the experiments you would do to test the properties of imaginary numbers (those expressed in terms of the square root of -1).

[HMS_Irruncible]

Some purists incorrectly insist on conflating “science” with the “scientific method.” They’re the only ones who would say that math isn’t a science. These same people would say that it isn’t real science to spend weeks wading around in mud counting fecal pellets to learn about the population and health of the marsh. I say that’s a bunch of crap.

[Orbifold]

@David Simmons

From the point of view of mathematics does it really matter whether or not its operations agree with the real world? That is of great importance to physical scientists, engineers and other technictians but from a strictly mathematical viewpoint who cares?

(raises hand)

I care. I view mathematics as being akin to poetry: it’s a language we use to describe reality. From that perspective, the question of whether mathematics correctly models reality is not just a practical question for scientists and engineers, but also an aesthetic question for mathematicians. There is aesthetic value to nonsense poetry, but in my view there is also value in poetry that teaches us something about reality. Same for mathematics.

There’s also the piddling issue of wanting to contribute to the greater progress of society and all that. If my mathematical theorems have positive real-world consequences, that’s no small thing. I’m not just in this field because I like thinking about right-angled hexagons and other weird stuff.

[Colibri]

@NattoGuy

Some purists incorrectly insist on conflating “science” with the “scientific method.” They’re the only ones who would say that math isn’t a science.

These “purists” include both most scientists and, it would seem, most mathematicians.

These same people would say that it isn’t real science to spend weeks wading around in mud counting fecal pellets to learn about the population and health of the marsh. I say that’s a bunch of crap.

The purists aren’t the ones who say it isn’t real science to wade around in the marsh (and I at times have been one of the marsh waders), unless you are perhaps restricting “purists” to particle physicists and molecular biologists.

[Mathochist]

@Gary T

Now look under “mathematics” – my dictionary starts the definition off with “the science of numbers…”

Obviously a dictionary not written by a topologist, a geometer, an algebraist, or even many analysts…

[Mathochist]

@FriendRob

I’d argue that math is not only a science, it’s an experimental science.

Think about geometry. The postulates are statements about points, straight lines, etc. But “point”, “line”, etc. are never defined. We have to rely on an intuitive notion of these items in order to proceed. We can only know if our intuitive notions and postulates are correct by applying them and checking if they agree with the “real world”.

GAH! This hasn’t been true in over a hundred years! The whole point of abstract geometry is that the terms aren’t defined, but are whatever fits the posited axiomatic relationships. As Hilbert said, “The theorems of geometry should hold true if one everywhere substitutes for ‘point’, ‘line’, and ‘plane’ the terms ‘table’, ‘chair’, ‘beer mug’.”

[Derleth]

NattoGuy, your post doesn’t indicate that you know what you are talking about. Science is defined by the scientific method, in that the scientific method establishes the notions of falsifiability and repeatability that are essential to doing actual science as opposed to theology or metaphysics. If you cannot falsify the hypothesis or repeat the experiment, you are not doing science.

As for wading around in marshes, science is all about exploring the real world. It’s about making observations and measurements, as opposed to the Platonic notion of trying to derive the ‘truth’ about the world via a process of deduction from first principles. If you aren’t inductively building theories from actual observations, you are not doing science.

Mathematics touches on science in two ways: It is a precise way to write down observations and it is a precise way to express the relationships between quantities, which is often a good way to make predictions about future events.

Mathematics, however, is neither inductive nor falsifiable. Mathematics does proceed via deduction from first principles, and there is no way to ‘disprove’ an axiom. You may reject an axiom, but your rejection does not invalidate the math that accepts it. Non-Euclidean geometry in no way damages or disparages Euclidean geometry.

(You can make errors in math, and your lapses of logic are indeed subject to being disproven. But there is no way for someone to invalidate a valid argument the way Einstein invalidated aspects of Newtonian physics.)

Mathematics, being a language, is always infinitely precise. There is no error of measurement or smallest indivisible unit or uncertainty when dealing with mathematical structures, which need not be numbers. Science is never infinitely precise, because measurements are subject to the confounding effects of our tools being made of matter. Every measurement comes with a margin of error, a bound beyond which it is meaningless to carry on the decimal. Numbers have no such limits.

David Simmons: Ideally, no. It doesn’t make any difference, Orbifold’s aesthetics notwithstanding. In fact, it’s always surprised me just how well math does correspond to how the physical world works. And I’m not alone in this.

FriendRob, in any axiomatic system you have terms that aren’t defined simply because you need to define things in terms of them. In geometry, you have one set of undefined terms, in arithmetic you have another, and in topology you have a third.

[Mathochist]

@Derleth

In fact, it’s always surprised me just how well math does correspond to how the physical world works. And I’m not alone in this.

The commonly accepted term for this is “unreasonable effectiveness”. It’s really the only standing problem (IMHO) in philosophy of mathematics.

[Derleth]

@Mathochist

The commonly accepted term for this is “unreasonable effectiveness”. It’s really the only standing problem (IMHO) in philosophy of mathematics.

Is the article I linked to the origin of this term?