Mathematics as an empirical science

[wofsy]

I question whether mathematics is not an empirical science. True it has the power of absolute proof which is denied to Physics and Biology but except for that maybe it is the same.

Mathematicians look at "empirical data" which for them are mathematical objects - say such as Riemann surfaces - They examine the data through examples to understand their properties. They make hypotheses based upon these examinations and then test them on other examples. This may verify the hypothesis, or show that the hypothesis needs modification. At some point, if the investigation bears fruit, a theory may emerge that organizes the data and predicts the properties of yet unexamined data.

In short, it seems that mathematical research includes the same processes that are found in other sciences. Functionally, it is much the same.

 

[JoeDawg]

In short, it seems that mathematical research includes the same processes that are found in other sciences. Functionally, it is much the same.

To a certain degree, you can apply scientific ideas to math. You can, using logic, deduce or discover, the implications of a formula or axiom. However mathematics is essentially definitional, so its more a matter of delving into the contraints of your assumptions, than understanding the universe itself.

One could, in a similar way, apply scientific principles to the study of literature.

 

[madness]

I would say that the definitions are often motivated empirically. I would also say that our logic is derived empirically. So in this way I think mathematics is an empirical science.

 

[JSuarez]

However mathematics is essentially definitional,

I disagree; in most cases, mathematical objects are given definitions only after they crop up enough times, often in distinct forms and in different areas; only then, someone proposes to unify these seemingly disparate things under a common definition. Then a second phase starts: exploring the deductive consequences of the general definition, to see what more can gleaned from it. A typical example is Group Theory: it took centuries, with groups cropping up everywhere, before a formal definition was proposed, and many other examples exist.

I would say that the definitions are often motivated empirically. I would also say that our logic is derived empirically. So in this way I think mathematics is an empirical science.

I wouldn't go that far yet, but I'm also convinced that there is an empirical element. Even the correct forms of logical argumentation seem to be a product of natural selection, acting in our brains.

 

[rasmhop]

In short, it seems that mathematical research includes the same processes that are found in other sciences. Functionally, it is much the same.

The only reason why part of it is the same is that both the sciences and mathematics tries to gain insight and understanding. I doubt you could find a single academic discipline that doesn't share the properties you mentioned. This is just the way humans gain understanding; by experimenting with concrete objects, by trial and error, and by building on previous results. So all pursuits to gain understanding of something will include these elements.

If you really took the analogy seriously, then math would be extremely bad science because it doesn't keep re-evaluating its hypotheses. The main difference between math and the other sciences is that in the other sciences you can never be sure about the truth of a theory so you keep re-examining it and trying it in new contexts. In math you try it with maybe a few objects, you then prove it to hold in general, and then you don't need to re-examine the truth of the statement.

I wouldn't call math empirical, but rather say that a common approach to discovering truth is experimental.

 

[wofsy]

The Wikipedia article "Mathematics" has a section on Mathematics as Science.

It refers to results in modern philosophy and logic that I do not understand.

 

[JSuarez]

The only reason why part of it is the same is that both the sciences and mathematics tries to gain insight and understanding. I doubt you could find a single academic discipline that doesn't share the properties you mentioned. This is just the way humans gain understanding; by experimenting with concrete objects, by trial and error, and by building on previous results. So all pursuits to gain understanding of something will include these elements.

I agree, there is a unity in the method.

If you really took the analogy seriously, then math would be extremely bad science because it doesn't keep re-evaluating its hypotheses. The main difference between math and the other sciences is that in the other sciences you can never be sure about the truth of a theory so you keep re-examining it and trying it in new contexts. In math you try it with maybe a few objects, you then prove it to hold in general, and then you don't need to re-examine the truth of the statement.

Here I disagree, mathematics does re-evaluate its content (hypotheses is a tricky word: what is an hypotheses in Mathematics? The statement of a theorem, the validity of the logical steps in the proof(s), the nature or accepted definition of the concepts involved? There are several possibilities). This happens when someone proposes a new proof that is judged to be more rigorous than the ones before (even "rigour" is a complex concept, that evolves with time); for example, there proofs in Euclid that are still deemed rigorous today, but Analysis and Topology, for example, only attained a "rigorous" status well into the XXth century; granted that they appeared much later, but the concept, say, "function" is still evolving.
In the Natural Sciences (I don't like the name; I'm just using it to distinguish them from Mathematics), there are indeed theories that are revised (and even abandoned) when the range of empirical data widens and falsifies them. This is not so evident in Mathematics, but happens, for example, in the calls of the constructivist schools to take a more critical approach to the validity of the logical steps, or the proper format of proofs; the mais difference is that, in the NS, eventually everyone is forced to accept the evidence, while in Mathematics this is more subtle; one extreme example is Brouwer's model of the real continuum: almost nobody uses is, not because there is something intrinsically wrong with (in fact, its consistent iff the classical model is), but because there is no compelling reason (comparable to the pressure of evidence) to overhaul a quite large part of Mathematics in favour of the other.
Regarding truth, it's not a mathematical concept (but I believe that it's an objective one; I'm no relativist), but a metaphysical one, and we do not have a really good formal theory of it; it's perfectly possible that the truth of some statements, even mathematical ones, must be revised in the future. Other recent questions are the rise of experimental mathematics and the possible limitations to our hability of to prove some statements implied by Complexity Theory.

It refers to results in modern philosophy and logic that I do not understand.

This article is not very good. What parts are you referring to? The ones referring to 1930's are the failure of Frege and Russell's project to reduce Mathematics to Logic (I'm not talking about Gödel's Incompletness Theorems, please don't drag them here; they are not relevant to this). This reduction is currently being investigated again, with some surprising results: it seems that we need veru little beyond Logic to reconstruct most Mathematics.

 

[JoeDawg]

I disagree; in most cases, mathematical objects are given definitions only after they crop up enough times, often in distinct forms and in different areas; only then, someone proposes to unify these seemingly disparate things under a common definition.

Crop up where?

Mathematics certainly gets its starting point from empirical observation, and if you create a mathematical model of something, you can then go back and compare later on. And if the universe is consistent, then mathematics will reflect this consistency.

But that's somewhat different from saying that it is scientific. Science in large part relies on empirical observation, mathematics relies more heavily on logical deduction, as mathematics can work quite well, even if it has no empirical parallels. Its more useful of course, when it does.

 

[wofsy]

Crop up where?

Mathematics certainly gets its starting point from empirical observation, and if you create a mathematical model of something, you can then go back and compare later on. And if the universe is consistent, then mathematics will reflect this consistency.

But that's somewhat different from saying that it is scientific. Science in large part relies on empirical observation, mathematics relies more heavily on logical deduction, as mathematics can work quite well, even if it has no empirical parallels. Its more useful of course, when it does.

While I see your point. my experience is that mathematics begins with empirical observation of mathematical objects. A grand example is the theory of elliptic integrals.

Further, Mathematics is empirically studied through observations of the observed world. Many theorems and ideas of mathematics are visualized in natural phenomena.

One can test whether a mathematical idea is true through empirical observation. For instance, the Dirichlet principle was suggested in part by the convergence to a constant temperature distribution of a heated surface whose boundary temperature is held constant

 

[octelcogopod]

Maybe it is true that there cannot exist an object or phenomena in reality that cannot have a mathematical model, but there can exist a mathematical model that does not exist in reality.

This does not equate math and reality.. Quantification is obviously an important part of reality, but the map is not the territory, as they say.
Not to mention, reality is extremely complex. A mathematical model may not have taken everything into account.

 

[wofsy]

This does not equate math and reality.. Quantification is obviously an important part of reality, but the map is not the territory, as they say.
Not to mention, reality is extremely complex. A mathematical model may not have taken everything into account.

Right - but no theory takes everything into account. Incompleteness does not make it non-scientific.

No mathematical theory accounts for all of mathematical objects just as no biological theory accounts for all of life.

 

[JoeDawg]

No mathematical theory accounts for all of mathematical objects just as no biological theory accounts for all of life.

But regardless of theory, mathematical objects are entirely created by the mind, whereas empirical objects have an external source. So you're not really 'observing' mathematical objects the way you observe empirical ones.

Its similar to the idea of 'nothing'. Its just a negation. There is no thing, called nothing. But language allows us to treat 'nothing' as if it existed. This is a useful contradiction.

 

[Pythagorean]

Mathematics is a diverse academic field.

In some ways, we can say that it's postulates were invented, and mathematics is now the empirical study of what those postulates mean.

In other ways mathematical discovery and invention were the direct result of physical observations (Geometry and the planets, for instance).

Mathematics is more generalized than physical science; That is, we can use mathematics to simulate a universe with different parameters. Or another example is complex numbers. In physics, we generally ignore the imaginary parts, or resolve them to get real results.

We get negative frequencies from formal mathematics, but we don't define frequency that way in physics, so the negative frequencies are equivalent to the positive frequencies (which presents conceptual pitfalls when comparing a double-sided spectrum to a single-sided spectrum because the negative frequencies can be shown to contribute to power mathematically.)

 

[JoeDawg]

In some ways, we can say that it's postulates were invented, and mathematics is now the empirical study of what those postulates mean.

I agree with this.

In other ways mathematical discovery and invention were the direct result of physical observations (Geometry and the planets, for instance).

Not so much with this. Certainly the physical observations were not 'invented' but the math to describe them is.

Its like if you observe a squirrel collecting nuts, you can describe this as a strategy for survival, even turn it into an equation about survival, but that equation is just a description, your description can be as detailed or as general as it is useful to you. And you could describe it different ways, depending on what you see as important.

 

[Pythagorean]

I agree with this.

Not so much with this. Certainly the physical observations were not 'invented' but the math to describe them is.

Its like if you observe a squirrel collecting nuts, you can describe this as a strategy for survival, even turn it into an equation about survival, but that equation is just a description, your description can be as detailed or as general as it is useful to you. And you could describe it different ways, depending on what you see as important.

I don't see the conflict:

"mathematical discovery and invention were the direct result of physical observations"

In other words, planetary orbits were observed and this lead to geometry. I.e. a part of math was invented from physical observations. I'm not claiming that geometry and planetary orbit are the same thing, but that the invention of geometry was a direct result of observing physical shapes.

You can also further investigate the physical observation to discover more about the mathematics. (Likewise, investigating the mathematical properties can lead to more insight on the physical observation). The relationship between mathematics and physical science have always been complementary in this manner.

 

[JoeDawg]

In other words, planetary orbits were observed and this lead to geometry. I.e. a part of math was invented from physical observations. I'm not claiming that geometry and planetary orbit are the same thing, but that the invention of geometry was a direct result of observing physical shapes.

Oh, ok, absolutely, it seemed as though you were implying some kind of inevitablity. But I agree with what you have said here.

 

[wofsy]

Oh, ok, absolutely, it seemed as though you were implying some kind of inevitablity. But I agree with what you have said here.

But more - theorems of mathematics are experimentally suggested by natural phenomena e.g. the Dirichlet principle. this is not the same as deduction from axioms. It is induction from experience.

 

[JoeDawg]

theorems of mathematics are experimentally suggested by natural phenomena

If you mean you can do experiments and then create formulas to describe them, sure, but I've never met an experiment that could speak.

 

[wofsy]

If you mean you can do experiments and then create formulas to describe them, sure, but I've never met an experiment that could speak.

I mean that observation suggests mathematical theorems. Creating formulas is not what mathematics is about.

The case of the Dirichlet Principle well represents what I mean. It would be helpful for me to go through it to see why it's discovery is not an example of inductive reasoning from experiment.

 

[JoeDawg]

I mean that observation suggests mathematical theorems.

Mathematicians suggest mathematical theorems, based on inductive and/or deductive logic.

Giving 'observation' an anthropomorphic nature does nothing, but take away credit from those who create the theorems.

 

[wofsy]

Mathematicians suggest mathematical theorems, based on inductive and/or deductive logic.

Giving 'observation' an anthropomorphic nature does nothing, but take away credit from those who create the theorems.

I don't understand

 

[JoeDawg]

I don't understand

An observation doesn't suggest anything. People can find patterns in observation, and draw inspiration from observation, but I guarantee someone not versed in mathematics... will not hear what your observation is 'suggesting'. That makes no sense.

Its not the observation that is important, its what the person does, and is capable of doing with it that counts.

 

[wofsy]

An observation doesn't suggest anything. People can find patterns in observation, and draw inspiration from observation, but I guarantee someone not versed in mathematics... will not hear what your observation is 'suggesting'. That makes no sense.

Its not the observation that is important, its what the person does, and is capable of doing with it that counts.

yeah but what does that have to do with this discussion. What you say is true of any science not to mention all experience.

 

[JoeDawg]

yeah but what does that have to do with this discussion. What you say is true of any science not to mention all experience.

Well then I don't understand your 'but more...'

Two apples suggests 1+1=2.
I don't see any special relevance to that.
That's not science, that is just observation.

How are your theorems different?

 

[Tinyboss]

Yes, a mathematician attempting to prove a conjecture will often begin by looking at examples. If one of them turns out to be a counterexample, then that's that--time to quit or revise the conjecture. But if every single example she checks fits the conjecture, it means nothing. The next one could turn out to be false (except in the scenario where there are a finite number of cases to check--but these are often considered to be "ugly" proofs e.g. the four color theorem).

In exceptional cases (the Riemann Hypothesis is the standard example), mathematicians may feel confident that a conjecture is true, but be unable to prove it. Many, many results in number theory begin with, "If the RH is true, then ..." But such results DO NOT have the status of theorems. They are conditional...or looking at it in another way, they are theorems in the standard system with an additional axiom stating that the RH is true.

Other sciences, even rigorous ones like physics, don't have axioms or theorems--everything is conjecture. It makes sense, because math is the only science that doesn't have to concern itself with the real world. Often the real world is the inspiration for mathematical objects and results, but once these are formulated in mathematical terms, they cease to have anything to do with the physical phenomena from which they arose.