The Philosophy of Mathematics

[sofiarune]

So I was watching the discussion between Richard Dawkins and Neil deGrasse Tyson today and a comment by Tyson got me thinking... which one of the philosophies of mathematics is currently the most widely held and why?

Any math nerds out there want to help me out?

 

[DepricatedZero]

I THINK what you're looking for might be Newton's Principia? It outlines Mathematics in Nature - actually a quick google search shows that the title translates to "Mathematical Principles of Natural Philosophy." Is this how you mean?

 

[lrkun]

I'm not familiar with the philosophy of mathematics. This is actually the first time I've heard about it. What's your take on this?

 

[Master_Ghost_Knight]

There is no seperate entities of the "philosophy" of mathematics. Even tough you can sort of recognise some common themes and decide to call that a group, like algebra, geometry among others, but there are many cross results that come from "one field" to be used on "another", far more so then what it hapears from the surface (we wouldn't have 1% of the knowledge we have today about math if that wasn't so) and certainly we wouldn't have the parts that we deam usefull if not for the others that don't appear to have any real aplication.
However, there is a field that is more directly usefull caled "parcial differential equations" that use to model almost every dynamic system (but to solve that you will have to know, matrix operations, transformations, fields among others).

 

[sofiarune]

Thanks for the replies guys but I'm not sure you realize what I meant. I literally mean the philosophy of mathematics, particularly contemporary schools of thought. You can learn more about them here http://en.wikipedia.org/wiki/Philosophy_of_mathematics . It's not so much distinguishing different types of math or finding new ways to use it, it's trying to answer what math is in a broad context. Think of it as analagous to the philosophy of science.

SO, now that that's out of the way. Anyone know?

 

[Master_Ghost_Knight]

Oh the philosophical undertone of math and not math as a philosophy. Great, that is easy, logic is by far the most important in that aspect, no doubt about it.

The empericist view (any of them plus socioconstructivism) is dead wrong, there is no such thing in mathematical fomalization.

 

[ImprobableJoe]

I think philosophy is mostly a giant load of BS created by people who refuse to get real jobs.

 

[Master_Ghost_Knight]

I think philosophy is mostly a giant load of BS created by people who refuse to get real jobs.

Then you don't know philoophy. Although I give it to you that there isn't much to be done in the developmet of threads of reasoning, but anyways I still believe that if people were more philosophycal savy (which most people don't have the slightest clue of what is about much less use it) then there wouldn't be loads of the bullshit we see today. Note: topic best fit in philosophy.

 

[ImprobableJoe]

Then you don't know philoophy. Although I give it to you that there isn't much to be done in the developmet of threads of reasoning, but anyways I still believe that if people were more philosophycal savy (which most people don't have the slightest clue of what is about much less use it) then there wouldn't be loads of the bullshit we see today. Note: topic best fit in philosophy.

We need to keep philosophy away from useful things like science and mathematics, so they don't screw it up.

 

[Master_Ghost_Knight]

We need to keep philosophy away from useful things like science and mathematics, so they don't screw it up.

It may seam that way but we are a bit to late. Philosophy has established mathematics and science almost as a seperate field, it was due to the development of the "good pactices of reasoning" (that requiered logic to develop) that shaped the basic characteristics of both science and math (like objectivity, formalism), math and science went their seperate ways because math can be done on logic alone while science requiers an acessment of reality (empiricism) to establish reality. There is allot of worthy knowledge of philosophy on how to establish a proper argument, how to think, the logical falacies is a subject that comes from philosophy (particularly related to the topic of logic, duh), in a sense there is merit there. In fact in early history science and math were undistinguishable from philosophy. However that usefull part is somewhat distinct from the retarded ofspring as the epistemics of the conteporaneous socio-economic conundrums (or insanities alike). It is a just a deseased field of knowledge plagued with crap, but it is not all bad reasoning... just most of it.

 

[DepricatedZero]

I think the Principia still fits then, since it addresses how math is simply a description of nature

 

[sofiarune]

I think the Principia still fits then, since it addresses how math is simply a description of nature

Yes but is that the predominant school of thought right now? I'm inclined to go the mathematical realist route but was hoping a math geek could enlighten me none the less.

 

[fzwl]

When I'm doing mathematics, I behave as if I'm a Platonist. Most of the time I don't think about what philosophy I subscribe to. If I do think about it, then formalism seems like the only easy to justify philosophy. (Platonism seems unbelievable - but only when I think about it.) I don't like the social philosophies, although some of the folks who subscribe to those philosophies make some good points.

I don't know what most other mathematicians think, but I suspect my view is fairly common.

 

[Nelson]

After skimming the wiki page, I believe I would fall into the category of Logicist. However, I am a physicist/astronomer, and while I have spent a good bit of time pondering the philosophy of science, I haven't spent much time at all thinking about the philosophy of mathematics in particular. I mostly view mathematics as a means to an end (understanding/modeling physical reality). I suppose my position could be easily swayed by someone who is more experienced with the field of pure mathematics and presents a convincing argument for why mathematics isn't simply an extension of logic.

Can anyone present an example of a mathematical concept or theorem that does not conform to the following?

Rudolf Carnap (1931) presents the logicist thesis in two parts:

1. The concepts of mathematics can be derived from logical concepts through explicit definitions.
2. The theorems of mathematics can be derived from logical axioms through purely logical deduction.

then formalism seems like the only easy to justify philosophy

I suppose I also agree with this a bit, but Formalism seems to supersede Logicism as it pertains to both logic and mathematics. These two ideas don't seem to be mutually exclusive.

 

[Andiferous]

There is allot of worthy knowledge of philosophy on how to establish a proper argument, how to think, the logical falacies is a subject that comes from philosophy (particularly related to the topic of logic, duh), in a sense there is merit there. In fact in early history science and math were undistinguishable from philosophy. However that usefull part is somewhat distinct from the retarded ofspring as the epistemics of the conteporaneous socio-economic conundrums (or insanities alike). It is a just a deseased field of knowledge plagued with crap, but it is not all bad reasoning... just most of it.

:facepalm:

Are you going into formal logic there, deer?

 

[Zetetic]

Oh the philosophical undertone of math and not math as a philosophy. Great, that is easy, logic is by far the most important in that aspect, no doubt about it.

The empericist view (any of them plus socioconstructivism) is dead wrong, there is no such thing in mathematical fomalization.

Are you talking about the finitist view? If so I agree that it is de facto false, given that it is useful to be able to take in to account circumstances that have a logical form requiring infinite.

I think that New Empiricism is generallycorrect http://en.wikipedia.org/wiki/Philosophy_of_mathematics#New_Empiricism
In order to argue against it you would have to be a dualist of some sort, which I find to be a silly and confusion position.

I don't know that New Empiricism gets the whole picture, but it does understand that there is necessarily a physical engine (a human brain) capable of organizing it's perceptual patterns and generalizing them to apply to predicted outcomes.

Caveat before reading this paragraph: (I think that social constructivists who insist that a mathematical theory can be discarded do not understand mathematics)
There is necessarily -some- sociological element to mathematics, to say otherwise seems wholly ignorant of the way mathematics develops. I am more likely to work on a problem if it is generating buzz in the mathematical community, but I am also more likely to work on a problem that relates to a system of mathematics previously specified. It is not very common for someone to simply build an abstract mathematical formalism from scratch with no reference to other mathematical ideas. Therefore there are absolutely sociological aspects to the progression of mathematics (though the affect on actual mathematical processing is unclear at best).

Mathematical Platonism is so totally muddled as to be useless (it cannot define what existence in the abstract means in any concrete way).

Formalism gives an incomplete picture, though I consider it generally correct despite this. They fail to take in to account other mathematical intuitions, however. You must also consider formal games with mental images and the like in general to be considering the process of doing mathematics. I think that it is likely that some mathematicians do not visualize anything but string manipulation. One of my professors studied under Kleene and insists that Kleene thought in this way almost exclusively.

My position is the following: Mathematical formalisms are methods of organization, either of other mathematical objects or of perceptual phenomena. So numbers were originally a method of organizing the perceptual phenomenon of keeping track of goods and so forth. Then number theory is the method of organizing and discovering facts about the number system.

The social aspect comes in in the notational conventions and in what direction mathematical research takes at a given time. In this way, there is a sociology of mathematics, but it is not related to how the individual problems are solved (though the difficulty and prestige can be a source of sociological input).

EDIT: this is arguably a form of realism since I hold that any brain with the ability to organize it's perceptions would be able to develop mathematics.

 

[Zetetic]

After skimming the wiki page, I believe I would fall into the category of Logicist. However, I am a physicist/astronomer, and while I have spent a good bit of time pondering the philosophy of science, I haven't spent much time at all thinking about the philosophy of mathematics in particular. I mostly view mathematics as a means to an end (understanding/modeling physical reality). I suppose my position could be easily swayed by someone who is more experienced with the field of pure mathematics and presents a convincing argument for why mathematics isn't simply an extension of logic.

Can anyone present an example of a mathematical concept or theorem that does not conform to the following?

Rudolf Carnap (1931) presents the logicist thesis in two parts:

1. The concepts of mathematics can be derived from logical concepts through explicit definitions.
2. The theorems of mathematics can be derived from logical axioms through purely logical deduction.

Was Carnap referring to predicate logic? I assume he was considering that he was a logical positivist. The second assertion might arguably violate Godel's first incompleteness theorem. The question I would ask Carnap (were he not dead) is whether he is claiming here that mathematics reduces to a single axiom system? If he considers this to be the case, he may be wrong about 2.

EDIT: I also should point out that the lines between logic and mathematics have become fairly blurry if you consider fuzzy logic, modal logics etc. to be part of logic proper.

 

[ImprobableJoe]

It may seam that way but we are a bit to late. Philosophy has established mathematics and science almost as a seperate field, it was due to the development of the "good pactices of reasoning" (that requiered logic to develop) that shaped the basic characteristics of both science and math (like objectivity, formalism), math and science went their seperate ways because math can be done on logic alone while science requiers an acessment of reality (empiricism) to establish reality. There is allot of worthy knowledge of philosophy on how to establish a proper argument, how to think, the logical falacies is a subject that comes from philosophy (particularly related to the topic of logic, duh), in a sense there is merit there. In fact in early history science and math were undistinguishable from philosophy. However that usefull part is somewhat distinct from the retarded ofspring as the epistemics of the conteporaneous socio-economic conundrums (or insanities alike). It is a just a deseased field of knowledge plagued with crap, but it is not all bad reasoning... just most of it.

In another sense, it could be compared to... ummm... let's say "carriages and cars" as an example.

The horse-drawn carriage led to weird steam-powered buggies, that led to the Model T, that led through a whole bunch of steps to the Lamborghini Murcielago. Whatever was useful from the horse-drawn carriage has already been stripped from the useless, and is currently being applied wherever it still serves a purpose. It is possible that some of the lessons of the horse-drawn carriage have historical interest, and can serve as a basis for education in certain fields, the way that Newtonian physics are a useful but insufficient component of an education in modern physics.

On the other hand, there's not much modern use for historical carriage production, and it is not relevant to modern life. I doubt anyone feels the need to examine carriage design before creating a new high-end sports car, or any other vehicle. And I don't think there's really much in the way of new reseach in Newtonian physics, is there?

In the same way, there's not much new and of practical use in philosophy IMO. It has historical value, and it is probably useful to study what has already been done, but I'm not sure any significant amount of active philosophers are doing anything new or interesting that has any real relevance to modern life.

 

[Master_Ghost_Knight]

Are you talking about the finitist view?

No.

I think that New Empiricism is generallycorrect
In order to argue against it you would have to be a dualist of some sort, which I find to be a silly and confusion position.

I would say it depends of what you mean by dualism. Mathematics is conceptual in nature, that doesn't mean that is physically composed out of ethereal elements.

I don't know that New Empiricism gets the whole picture, but it does understand that there is necessarily a physical engine (a human brain) capable of organizing it's perceptual patterns and generalizing them to apply to predicted outcomes.

New empiricism is empiricism, meaning that is based on the subjective experience of the senses, meaning that we do math by like 5 yo, i.e. putting 1 apple with another apple getting 2 apples meaning that 1+1=2. If you have actually done math, you would know that whatever happens in the real is completely irrelevant to the propositions of math. If for instance every time you put together 1 apple with another apple in a box and you get 3 apples, 1+1 would still be 2 regardless of the physical experience that you perceive to be addition.

There is necessarily -some- sociological element to mathematics, to say otherwise seems wholly ignorant of the way mathematics develops. I am more likely to work on a problem if it is generating buzz in the mathematical community, but I am also more likely to work on a problem that relates to a system of mathematics previously specified. It is not very common for someone to simply build an abstract mathematical formalism from scratch with no reference to other mathematical ideas. Therefore there are absolutely sociological aspects to the progression of mathematics (though the affect on actual mathematical processing is unclear at best).

In a sense there is a social pressure to the development of mathematics in the sense that there need to be people subjected to social circumstances discovering its implications. But the propositions found in math is completely independent of any socio-economic circumstance, the fact that there is one and only one decomposition in prime factors of any positive integer would still be true regardless for instance if you believe in Jesus or not.

Mathematical Platonism is so totally muddled as to be useless (it cannot define what existence in the abstract means in any concrete way).

Sorry to tell you this, but it is not only useful as it is also the only way we do math, if it is not abstract it is not math period (it is one of the basic properties that makes math math).

Formalism gives an incomplete picture, though I consider it generally correct despite this. They fail to take in to account other mathematical intuitions, however. You must also consider formal games with mental images and the like in general to be considering the process of doing mathematics. I think that it is likely that some mathematicians do not visualize anything but string manipulation. One of my professors studied under Kleene and insists that Kleene thought in this way almost exclusively.

Again, no. It is not just generally correct, nothing in mathematics is established unless it is formal, you can dwell in "informal" ideas in order to plan a line of reasoning to follow but you will never establish anything unless you "walk the formal path".
So in essence your ideas on mathematics are completely upside down.

 

[Master_Ghost_Knight]

In another sense, it could be compared to... ummm... let's say "carriages and cars" as an example.
The horse-drawn carriage led to weird steam-powered buggies, that led to the Model T, that led through a whole bunch of steps to the Lamborghini Murcielago. Whatever was useful from the horse-drawn carriage has already been stripped from the useless, and is currently being applied wherever it still serves a purpose. It is possible that some of the lessons of the horse-drawn carriage have historical interest, and can serve as a basis for education in certain fields, the way that Newtonian physics are a useful but insufficient component of an education in modern physics.
On the other hand, there's not much modern use for historical carriage production, and it is not relevant to modern life. I doubt anyone feels the need to examine carriage design before creating a new high-end sports car, or any other vehicle. And I don't think there's really much in the way of new reseach in Newtonian physics, is there?
In the same way, there's not much new and of practical use in philosophy IMO. It has historical value, and it is probably useful to study what has already been done, but I'm not sure any significant amount of active philosophers are doing anything new or interesting that has any real relevance to modern life.

No, it is better compared to a computer. You have your hardware and you want to run a MatLab program, if you try to run that program directly on the hardware you will fail miserably. To succeed you will first have to install an operating system (which is internally divided in several layers) which is going to manage your CPU, memory and whatever, then on top you have to install MatLab. Even though MatLab requires the CPU to crunch numbers and memory to save states, it does not do so directly with the hardware; it interacts with operating system to access the machine resources. Then on top of MatLab you do your math coding, the code is given to the MatLab to be interpreted and the MatLab interacts with the OS to give the necessary resources to complete the task. Even though you don't need to worry anymore how to work with the hardware (how to allocate memory or manage the CPU) or internally how MatLab works it doesn't mean that the Operating System or MatLab is now useless.
Also there is nothing that stops me from adding new Hardware that I want to run with my Mat program, but to do that now I have to worry how to operate with the hardware, how to change the operating system to accommodate my new hardware, how MatLab works in order to make it call upon those new resources and only then do the coding.
And every day there are new computers (i.e. people) coming out which all require to you put on it an OS and then MatLab in order to be able to run your code.
Now that doesn't mean that all functions of MatLab are usefull, I may see it as a completly irrelevant the function of being able to estimate the number of toasts you eat a day.