Some questions about what math ##is## in general

[STEMucator]

What are theorems (lemmas, axioms, etc) exactly?

I know what a theorem is, and I know how theorems are motivated; I also know what lemmas and axioms are.

My question concerns something a little deeper.

Are these theorems and mathematical tools we have merely a consequence of the way we've defined things?

Are they discovered or created?

Feel free to share some thoughts.

 

[HallsofIvy]

What are theorems (lemmas, axioms, etc) exactly?

I know what a theorem is, and I know how theorems are motivated; I also know what lemmas and axioms are.

My question concerns something a little deeper.

Are these theorems and mathematical tools we have merely a consequence of the way we've defined things?

Well, the way we have defined things and the axioms we state using those definitions. We can have a mathematical structure in which the statement "through any point, not on a given line, there exist exactly one line parallel to the given line" is true and another in which it is false.

Are they discovered or created?

Feel free to share some thoughts.

Both! Theorems are created when we give basic definitions and axioms. We then dscover them when we prove them.

 

[jedishrfu]

I would have to say they are created. The earliest notions of math started out as collections of discoveries and later rules to transform results. Then as the need for a system of math, axioms were developed to define what can't be proved and to then carefully construct theorems from the axioms.

Euclid's geometry is perhaps the earliest and best example of a system of mathematics based on axioms and carefully constructed theorems. More recently, questions came up about the parallel axiom which resulted in the creation new geometries.

I imagine that if a different parallel axiom were chosen then perhaps spherical geometry would have been created before Euclidean geometry.

 

[lostcauses10x]

Created. What is a number but a symbol.

 

[STEMucator]

Well, the way we have defined things and the axioms we state using those definitions. We can have a mathematical structure in which the statement "through any point, not on a given line, there exist exactly one line parallel to the given line" is true and another in which it is false. Both! Theorems are created when we give basic definitions and axioms. We then dscover them when we prove them.

Why are axioms self evident truth? That's as if assuming they're absolute from the get go. Humans seem to have systematically created all of this "mathematics" at the end of the day.

For example, the way we define the addition operation (+). It means to literally take two precisely similar objects and combine them. "We" have defined it as such though, through... physical observation? Is math simply physics from the ground up then?

 

[Enigman]

Why are axioms self evident truth?

Axiom's don't have to be anywhere near the truth. They are just statements on which the structure is built and hence can't be proven using the structure.
For example the euclidean geometry with its axioms holds only to an approximation in General relativity. (http://www.thefullwiki.org/Non-Euclidean_geometry)

That's as if assuming they're absolute from the get go. Humans seem to have systematically created all of this "mathematics" at the end of the day.

For example, the way we define the addition operation (+). It means to literally take two precisely similar objects and combine them. "We" have defined it as such though, through... physical observation? Is math simply physics from the ground up then?

Mathematics at the end of the day is a tool (wow...I just spelled that troll...too much net...) to describe and solve real world problems (or was created to do so).

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself.
http://en.wikipedia.org/wiki/Mathem...ure_and_applied_mathematics.2C_and_aesthetics

An interesting read: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences

 

[STEMucator]

Axiom's don't have to be anywhere near the truth. They are just statements on which the structure is built and hence can't be proven using the structure.
For example the euclidean geometry with its axioms holds only to an approximation in General relativity. (http://www.thefullwiki.org/Non-Euclidean_geometry)Mathematics at the end of the day is a tool (wow...I just spelled that troll...too much net...) to describe and solve real world problems (or was created to do so). An interesting read: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences

I found this to be the most interesting portion of the darthmouth.edu article:

Mathematics does play, however, also a more sovereign role in physics. This was already implied in the statement, made when discussing the role of applied mathematics, that the laws of nature must have been formulated in the language of mathematics to be an object for the use of applied mathematics.

The statement that the laws of nature are written in the language of mathematics was properly made three hundred years ago;[8 It is attributed to Galileo] it is now more true than ever before. In order to show the importance which mathematical concepts possesses in the formulation of the laws of physics, let us recall, as an example, the axioms of quantum mechanics as formulated, explicitly, by the great physicist, Dirac.

There are two basic concepts in quantum mechanics: states and observables. The states are vectors in Hilbert space, the observables self-adjoint operators on these vectors. The possible values of the observations are the characteristic values of the operators but we had better stop here lest we engage in a listing of the mathematical concepts developed in the theory of linear operators.

It is true, of course, that physics chooses certain mathematical concepts for the formulation of the laws of nature, and surely only a fraction of all mathematical concepts is used in physics. It is true also that the concepts which were chosen were not selected arbitrarily from a listing of mathematical terms but were developed, in many if not most cases, independently by the physicist and recognized then as having been conceived before by the mathematician.

It is not true, however, as is so often stated, that this had to happen because mathematics uses the simplest possible concepts and these were bound to occur in any formalism. As we saw before, the concepts of mathematics are not chosen for their conceptual simplicity. Even sequences of pairs of numbers are far from being the simplest concepts, but for their amenability to clever manipulations and to striking, brilliant arguments.

Let us not forget that the Hilbert space of quantum mechanics is the complex Hilbert space, with a Hermitean scalar product. Surely to the un-preoccupied mind, complex numbers are far from natural or simple and they cannot be suggested by physical observations. Furthermore, the use of complex numbers is in this case not a calculational trick of applied mathematics but comes close to being a necessity in the formulation of the laws of quantum mechanics. Finally, it now begins to appear that not only complex numbers but so-called analytic functions are destined to play a decisive role in the formulation of quantum theory. I am referring to the rapidly developing theory of dispersion relations.

After reading, I find mathematics has become fine tuned to the standard model when concerning physics, but also has its own independent motivation. Mathematics is more of a formalism, whilst physics tends to be concerned more with phenomena.

Galileo's comment was striking, I'm not sure how he came to the conclusion or how he justified it though. Perhaps I should ask a philosopher.

 

[WannabeNewton]

It is what it is.

 

[1MileCrash]

Are these theorems and mathematical tools we have merely a consequence of the way we've defined things?

Yes. All mathematical truths come from our chosen set of axioms and conventions.

 

[elegysix]

Are these theorems and mathematical tools we have merely a consequence of the way we've defined things?

Yes. All mathematical truths come from our chosen set of axioms and conventions.

They are not just a consequence of the way we've defined things.
We've defined these theorems and tools so that they describe reality, and in that way these theorems are a consequence of how reality is.
We create the theorems and tools - but the obedience of reality to them is a discovery.

 

[pwsnafu]

When people talk about "describing reality" there are two ways this can be interpreted: describing the observable universe, and describing mathematical realities. Banach-Tarski for example describes the latter not the former.

 

[STEMucator]

They are not just a consequence of the way we've defined things.

We've defined these theorems and tools so that they describe reality, and in that way these theorems are a consequence of how reality is.

If "we've" defined these theorems and tools so that they describe reality, then clearly "we" are describing reality with our own perceptions (which happen to mostly be physical). So how can we say these theorems are a consequence of reality when it is our perception of reality that makes them real. Can you say with confidence that mathematics is completely objective in this sense? Probably not.

We create the theorems and tools - but the obedience of reality to them is a discovery.

I agree with this.

 

[HallsofIvy]

Why are axioms self evident truth?

You quote my previous post and then ask this? I didn't say that axioms are "self evident truths" because I don't believe any such thing! "Axioms" are part, along with definitions, of what determine the particular logic system we are working with.

That's as if assuming they're absolute from the get go. Humans seem to have systematically created all of this "mathematics" at the end of the day.

For example, the way we define the addition operation (+). It means to literally take two precisely similar objects and combine them. "We" have defined it as such though, through... physical observation? Is math simply physics from the ground up then?

I know many different ways that addition is defined in different systems. You are basing your questions on things that just aren't true.

 

[1MileCrash]

They are not just a consequence of the way we've defined things.
We've defined these theorems and tools so that they describe reality, and in that way these theorems are a consequence of how reality is.
We create the theorems and tools - but the obedience of reality to them is a discovery.

That any axioms or convention "describe reality" is still a choice on our part. We could take a new set of axioms that do not "describe reality" and derive new truths from them, that are just as true. Math is beyond "describing reality."

 

[R136a1]

That any axioms or convention "describe reality" is still a choice on our part. We could take a new set of axioms that do not "describe reality" and derive new truths from them, that are just as true. Math is beyond "describing reality."

Yes, it is a choice on our part. But I don't think that there's much interests in axioms that do not describe reality in some way or another. Virtually all the math you learn in university is invented in order to understand something better, and that something is usually very closely related to the reality.

Also, people get too hung up on the entire axiom process. Axioms are a way of presenting mathematics and to see whether everything is sound. It is not something to actually use in doing mathematics. Throughout history, you always see the same trend. Some mathematician does some research and finds some cool results. And only then does he think about which axioms he should use. People have worked with calculus for hundreds of years, and only very late in the game did they actually care about the specific axiom system. Same with groups, topology, differential geometry, etc.

Axioms are usually chosen to abstract something that is present in reality and intuitively true (intuitive means intuitive for the experienced mathematician). While some mathematicians do invent some completely unreal axiom systems, they are rarely useful and are not studied a lot for that reason.

Another source which shares my point of view on mathematics: http://pauli.uni-muenster.de/~munsteg/arnold.html

 

[1MileCrash]

All I'm saying is that there are no mathematical truths independent of the axioms we choose to adopt, this is the direct response to the question I quoted. You can discuss our motivations behind adopting certain axioms or whatever, but that doesn't have much to do with the question the poster asked.

 

[R136a1]

All I'm saying is that there are no mathematical truths independent of the axioms we choose to adopt, this is the direct response to the question I quoted. You can discuss our motivations behind adopting certain axioms or whatever, but that doesn't have much to do with the question the poster asked.

I think this is exactly what the poster is asking.

 

[1MileCrash]

Well, I don't see anything suggesting that the poster is asking why we adopt certain axioms, or anything about limiting math to the physical world. The questions are, if theorems and tools in math are consequences of how we've chosen to define things, and if these theorems and tools are discovered or created.

My opinion is that:
1.) yes, absolutely.
2.) foundations are created, we discover the results.

I don't see how pointing out that we've picked axioms that are consistent with human perception is some kind of disagreement to what I've said, nor do I see relevance to the poster's question. He's asking about mathematical philosophy, not about using it to solve human problems.

 

[Evo]

See https://www.physicsforums.com/showthread.php?t=645214 Thread closed.