Wake up, Math thread -

Testycalibrated · 07-21-2015, 09:38 PM

I think it's just the process of pattern recognition.

Heinz Hershold · 07-22-2015, 07:42 AM

:

in the spirit of reviving the math forum, here's a sloppy but hopefully deep question/thought

---------

it seems to me that many important contributions to math consist of taking an object that is hard to study, and building a user-friendly "scaffolding" around it, to enable you to make progress.

- some obvious examples are discretisation of functions, to approximate integrals or solutions of diff eqs. however, several other possible examples strike me:
- homology, to study homotopy theory
- hell, homotopy theory to study topological spaces
- matrix representations of groups
- differential and integral calculus do (or at least begin with) approximations of properties of functions by looking at "small" parts of the curve, at "every" place along the curve.
- Taylor polynomials replace a function with a polynomial that tries to mimic it
- similarly, Fourier series, etc.

is this a real pattern? and is there a term for it?

I may be misunderstanding what you are really thinking here, but I see all of these operations as algorithms; procedures used to simplify and solve specific types of problems but have contributed to the development of mathematics in general. For example, logarithms arose in the solution of compound interest calculations but soon found much wider applications throughout mathematics. In this instance of logarithms, it is easy to see how the transformation happened naturally as a method to solve a specific problem and the advantage it has in transforming multiplication and division into simpler operations of addition and subtraction would result in much wider use in other fields.

Some of the other transformations are not so easy or intuitive to see how they came to be developed, or even thought of in the first place. I can see how and why Heaviside developed operational calculus, in his work with electrical circuits with both transient and steady-state response, but it is not clear to me how or why Laplace developed his transforms, even though the two procedures are fundamentally the same.

I don’t think there is one process or pattern at work here that we can name and say that is what is behind all of these advances in mathematics. Sometimes these things just happen naturally in the course of trying to solve a problem, as it did with logarithms, and sometimes it comes about from some mathematical genius like Laplace doing whatever it is that a mathematical genius does with numbers for no particular reason, and sometimes a combination of the two, like Heaviside who was both a mathematical genius and a practical problem solver.

el jefe · 07-22-2015, 04:06 PM

:

I may be misunderstanding what you are really thinking here, but I see all of these operations as algorithms; procedures used to simplify and solve specific types of problems but have contributed to the development of mathematics in general. ...

i agree, though I am trying to say something more specific than that. they all involve approximating one thing with something else that has a more tractable structure.

Testycalibrated · 07-22-2015, 06:46 PM

Like freeman Dyson did with physics showing that the various quantum calculations are equivalent.

el jefe · 07-22-2015, 08:03 PM

I think you're thinking of dirac, but (I think) yeah

Requiem · 07-23-2015, 08:14 AM

:

I may be misunderstanding what you are really thinking here, but I see all of these operations as algorithms; procedures used to simplify and solve specific types of problems but have contributed to the development of mathematics in general. ...

i agree, though I am trying to say something more specific than that. they all involve approximating one thing with something else that has a more tractable structure.

I wonder whether the language of structuralism would help you here?

https://dl.dropboxusercontent.com/u/...%201997%29.pdf

Might give you some things to chew on.

el jefe · 07-23-2015, 11:53 AM

interesting, thank you. that will eat up a few bus rides to work...

Requiem · 07-23-2015, 12:02 PM

glad to be of service.

TestyCalibrate · 07-23-2015, 04:03 PM

:

I think you're thinking of dirac, but (I think) yeah

Nope. I'm thinking of dyson.

Heinz Hershold · 07-25-2015, 07:14 AM

:

interesting, thank you. that will eat up a few bus rides to work...

Interesting read, even though slightly over my head, but maybe it has given up the word you are looking for to answer your questions?

is this a real pattern?

and is there a term for it?

“structuralism”

And you don’t even need to read the whole paper to get that, just the title.

This paper may also have the answer to this other thread

Requiem · 07-25-2015, 07:21 AM

I'm not sure it provides a specific terminology or set of ideas which could be immediately translated to something which accounts for the kinds of activities that guapo identifies; but the whole "study of structures" thing is - to my mind at least - the kind of platitude that suggests an underlying notional possibility that might be able to account for them.

Heinz Hershold · 07-25-2015, 10:11 AM

:

I'm not sure it provides a specific terminology or set of ideas which could be immediately translated to something which accounts for the kinds of activities that guapo identifies; but the whole "study of structures" thing is - to my mind at least - the kind of platitude that suggests an underlying notional possibility that might be able to account for them.

Not to get into linguistic detours here, but I am curious about your use of the word platitude in that sentence.

Requiem · 07-25-2015, 11:48 AM

Good call, yeah in typical discourse a platitude is akin to a cliche or overused phrase with a moral underpinning, but in philosophy I've come across it being used to denote a basic statement about something that is not intended as a robust statement in itself, but more a starting point which one might, say, refine by argumentation in order to arrive at a more substantive definition.

"Maths is the science of structures" is not exactly a refutable statement as it stands, not without arguing over the meanings inherent thereto. In that way it's a platitude in the sense that it's "not really worth arguing over" without doing some real digging first. It's not meaningless, or cliched, but it is rather useless in terms of conveying any real substantive notions.

There are better terms I could have used but I'm afraid my brain was stuck on platitude.

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Messages:

07-22-2015, 08:03 PM	#2532215 / #30
el jefe Senior Member TR Pundit : Oct 2008 : 30,945	I think you're thinking of dirac, but (I think) yeah

07-23-2015, 11:53 AM	#2532599 / #32
el jefe Senior Member TR Pundit : Oct 2008 : 30,945	interesting, thank you. that will eat up a few bus rides to work...

07-23-2015, 12:02 PM	#2532603 / #33
Requiem Dubstyle. : Mar 2009 : i forget : 23,616	glad to be of service.

07-25-2015, 07:21 AM	#2533875 / #36
Requiem Dubstyle. : Mar 2009 : i forget : 23,616	I'm not sure it provides a specific terminology or set of ideas which could be immediately translated to something which accounts for the kinds of activities that guapo identifies; but the whole "study of structures" thing is - to my mind at least - the kind of platitude that suggests an underlying notional possibility that might be able to account for them.

07-25-2015, 11:48 AM	#2533891 / #38
Requiem Dubstyle. : Mar 2009 : i forget : 23,616	Good call, yeah in typical discourse a platitude is akin to a cliche or overused phrase with a moral underpinning, but in philosophy I've come across it being used to denote a basic statement about something that is not intended as a robust statement in itself, but more a starting point which one might, say, refine by argumentation in order to arrive at a more substantive definition. "Maths is the science of structures" is not exactly a refutable statement as it stands, not without arguing over the meanings inherent thereto. In that way it's a platitude in the sense that it's "not really worth arguing over" without doing some real digging first. It's not meaningless, or cliched, but it is rather useless in terms of conveying any real substantive notions. There are better terms I could have used but I'm afraid my brain was stuck on platitude.