Definition of mathematical object

[phoenixthoth]

definition of "mathematical object"

...And "mathematical existence." Do these phrases have accepted definitions? Back when Dedekind was rejecting Cantor's transfinite ideas, could there have been a definition Cantor would refer to and say definitively "my (infinite) sets have mathematical existence because the criteria of the definition of mathematical object are satisfied"?

There are many examples of mathematical objects. All structures are mathematical objects:
http://math.chapman.edu/cgi-bin/structures?HomePage

A proof is also a mathematical object.

Sets and categories are mathematical objects.

Any formal system is a mathematical object.

What is the common thread?

 

[phoenixthoth]

How about these for definitions:

Mathematical object:
That which is pointed to by a linguistic construct, known as an utterance, such that that utterance has at least one internally, logically-consistent interpretation.

Mathematical existence:
Something has mathematical existence if and only if it is a mathematical object.

 

[micromass]

Maybe you would have more luck on a philosophy forum. All of this is not really mathematics...

 

[phoenixthoth]

You're probably right except for the bit about this not being mathematics.

 

[micromass]

You're probably right except for the bit about this not being mathematics.

It's not mathematics. No single mathematician cares about these things unless they're somehow interested in philosophy.
I'm not saying your question is bad or boring. I'm just saying that it's not mathematics.

 

[reenmachine]

It's not mathematics. No single mathematician cares about these things unless they're somehow interested in philosophy.
I'm not saying your question is bad or boring. I'm just saying that it's not mathematics.

Can a mathematician be completely disinterested in philosophy? I'm not saying a mathematician will spend his time doing philosophy , but just to wonder what's going on at the most fundamental level of reality.

 

[phoenixthoth]

I would define mathematics as the study of mathematical objects. So a mathematician would care about what that means iff they care what mathematics is. Admittedly, some of us don't care but logicians for example who treat mathematics as a mathematical object must care.

This (the definition of mathematical object) is mathematics iff mathematics itself is a mathematical object and while I can understand your point of view, I disagree.

Now that that is out of the way perhaps we can work towards defining what a mathematical object is.

 

[Mark44]

Can a mathematician be completely disinterested in philosophy?

A mathematician can be interested in any number of things that aren't related to mathematics.

I'm not saying a mathematician will spend his time doing philosophy , but just to wonder what's going on at the most fundamental level of reality.

 

[phoenixthoth]

Can a mathematician be completely disinterested in philosophy? I'm not saying a mathematician will spend his time doing philosophy , but just to wonder what's going on at the most fundamental level of reality.

Some mathematicians think that the study of mathematics is what's going on at the most fundamental level of reality.

 

[jbriggs444]

Some mathematicians think that the study of mathematics is what's going on at the most fundamental level of reality.

Such an opinion would be a matter of philosophy. It has no mathematical content.

 

[micromass]

Some mathematicians think that the study of mathematics is what's going on at the most fundamental level of reality.

Who? I haven't met any mathematician who thinks that way.

 

[Tac-Tics]

@phoenixthoth - Mathematical objects are primitive notions of whatever formal system you happen to be using.

Most mathematicians ostensibly use something like set theory. To them, a mathematical object is anything that can be encoded as a set. (This includes pairs, relations, functions, integers, the reals, etc).

If I'm doing number theory, as the greeks might do, my objects would be just the integers.

I'm I'm doing category theory, my objects are categorical objects and morphisms.

If I'm doing type theory, my objects are types and terms and universes.

But there is no absolute notion of what an object is. In set theory, people don't generally consider "proofs" to be mathematical objects. In type theory, they are. Category theory is often built on top of some other formalization (again, often set theory), and you end up working with "two levels" of objects. The same is also true in ZF set theory, since ZF is built "on top of" first order logic. Meanwhile, type theory is built "on top of" the lambda calculus. Of course, I can define lambda calculus in terms of set theory, or set theory in terms of type theory.

In a way, it's very much the same as working with programming languages. Most languages have a notion of "object". But each language has their own definition. Most languages are turing complete and so (assuming the church-turing thesis) can bisumulate one another.

But again, there is no absolute notion. You can go on with your turtle-tossing all you want, but every time, there's another turtle sitting underneath.

 

[phoenixthoth]

Who? I haven't met any mathematician who thinks that way.

Sounds like a great topic for another thread.

If math isn't at all related to reality then it's basically the study of unicorns in the mathematician's basements; I refuse to believe that mathematics is not a study of the structure of reality.

 

[phoenixthoth]

But again, there is no absolute notion. You can go on with your turtle-tossing all you want, but every time, there's another turtle sitting underneath.

Yes and that's probably true of the process of defining anything: there are atomic words used in the definition. "There is no absolute notion," that is what I am trying to remedy...

 

[WannabeNewton]

http://forums.philosophyforums.com/logic-and-philosophy-of-math/
Enjoy

 

[phoenixthoth]

Thanks, though I was really hoping to hear others chime in on what is a mathematical object. I will try there, too

 

[chiro]

Most of mathematics looks at variation: in other words how can we express variation for particular kinds of objects.

This is expressed in the concept of a variable: it can be quite a range of structures including a numeric quantity, a map, a set, or something else but the key idea is that of variation.

By understanding variation, we seek to classify and analyze it in order to understand how it affects our understanding for when things change or behave differently.

This is true for everything from simple algebra to differential geometry: variation exists in all kinds of mathematical constructs.

If there was no need for variation or variability, mathematics would not need to exist.

 

[Fredrik]

I don't see the point of defining "mathematical object", especially if this is just supposed to be an odd way of specifying what mathematics is. I think it takes an essay to explain what mathematics is.

 

[Mark44]

Sounds like a great topic for another thread.

If math isn't at all related to reality then it's basically the study of unicorns in the mathematician's basements; I refuse to believe that mathematics is not a study of the structure of reality.

Much of mathematics is, or at least started out as something comparable to the study of unicorns. I doubt that many mathematicians would describe mathematics as studying the structure of reality.

Some examples.
Complex numbers, which have a real part and an imaginary part.
Vector spaces of dimension higher than 3, including infinite-dimension spaces.
Sphere packing in in fairly high-dimension spaces.

 

[pwsnafu]

Much of mathematics is, or at least started out as something comparable to the study of unicorns. I doubt that many mathematicians would describe mathematics as studying the structure of reality.

Some examples.
Complex numbers, which have a real part and an imaginary part.
Vector spaces of dimension higher than 3, including infinite-dimension spaces.
Sphere packing in in fairly high-dimension spaces.

Even better example is the Banach-Tarski theorem. In this example the unicorns in question are non-measurable sets.

 

[Uvohtufo]

We probably shouldn't seek a concise definition of 'mathematical objects', but in general I describe it by putting it in contrast with mathematical operations. If you are doing division for example, you divide one number by another, and you get an output. The inputs and outputs are objects. What you do with the inputs to get the outputs is an operation. What you operate with is your object.

So I don't consider a proof an object. The idea of a proof is too well defined. A proof is like giving new perspective between different parts of mathematics. You shed light on mathematical objects in relation to each other.

I think its kind of neat how mathematicians avoid and shun these kinds of questions. Doing math is for mathematicians, talking about math is for someone else.

 

[pwsnafu]

So I don't consider a proof an object. The idea of a proof is too well defined. A proof is like giving new perspective between different parts of mathematics. You shed light on mathematical objects in relation to each other.

Except in proof theory, we use mathematics to study proofs as objects .

 

[Uvohtufo]

Except in proof theory, we use mathematics to study proofs as objects .

Well that's ridiculous.

 

[HallsofIvy]

Even more ridiculous is "Category theory" where we have "objects" and "morphisms" with each morphism defined by an ordered pair of "objects". We have, for example, the "category of sets" in which the "objects" are sets and the morphisms are functions from one set to another. Or the "category of groups" in which the objects are groups and the morphisms are homomorphisms from one group to another.

 

[phoenixthoth]

Well that's ridiculous.

It's disheartening slightly that when we expand our understanding of what might a mathematical object be, there is resistance to this endeavor. This type of resistance has occurred many times in the history of math. What I find ridiculous is the resistance to opening more doors in terms of the study of the nature of mathematical objects.

Studying proofs as mathematical objects hints at what are known as formal systems. Now asking what a mathematical object is touches upon what the definition of math is. There are many different definitions of math, one I would possibly subscribe to is that math is the study of mathematical objects. Another definition of math could be that it is a particular formal system robust enough to describe all formal systems; math is, in a sense, both a formal system in its own right and the study of formal systems in general.

 

[Uvohtufo]

Wow, replying to me 4 months later. Thats cool.

Anyway, I would argue that as 'our understanding' expands, what we understand also changes, and it can hardly be claimed that we are understanding the same thing pre and post expansion. If you make a claim about some elaborated kind of proof, you can't infer anything about a proof was prior to your elaboration. Or better said, you are now using the word 'proof' in a different way than how you had been before you choose to elaborate into what a proof is.

And I would say the same about what Math is. We define the word math, not math itself, and no definition of the word will fully capture all the ways the word is used.

As a side note, I have since studied a bit of proof theory in the last 4 months, and I find it interesting.

 

[mathsman1963]

...And "mathematical existence." Do these phrases have accepted definitions? Back when Dedekind was rejecting Cantor's transfinite ideas, could there have been a definition Cantor would refer to and say definitively "my (infinite) sets have mathematical existence because the criteria of the definition of mathematical object are satisfied"?

There are many examples of mathematical objects. All structures are mathematical objects:
http://math.chapman.edu/cgi-bin/structures?HomePage

A proof is also a mathematical object.

Sets and categories are mathematical objects.

Any formal system is a mathematical object.

What is the common thread?

Not being of a philosophical disposition I would define a mathematical object as any object studied in a mathematical way by a mathematician.