The Philosophy of Mathematics

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On this view "pure" mathematics is an outgrowth from this practical process of developing methods of mathematical description, abstracting it to describe formalisms not yet having any application in the world other than to mathematical ideas themselves. If mathematics is a language, why should it have the precision it appears to do? Why should the universe be describable and comprehensible at all? If it is abstracted from the world why should we be able to develop it in non-world realms "mathematical ideas themselves" as you put it?

Why are we able to make abstractions, and in a precise way? I realise that in asking such questions I am in a way moving beyond your points and in some way beyond the point of the article. But I am trying to emphasise that in trying to "solve" this question, we employ assumptions that raise larger, wider questions.

This shouldn't be taken as an excuse not to raise such questions! Indeed, I was trying to inculcate a sense that there is still great mystery in the world. German, with its very specific words for very specific things and a high level of succinctness, seems much more precise. Could it be that mathematics as a language is just the language that is the most precise that we know of currently?

It is possible that math is "just" the most-precise currently-known language. But this doesn't answer any of the questions we have been raising, although it does possibly reframe the overall question to be something more like "how accurately are we able to describe patterns and is math the optimal language for so doing? I also note that precision is something, but not everything. In human languages, it is possible to convey things by ambiguity and choice of words that it is impossible to encode formally in precise language.

Understanding what "Feeling under a cloud" means takes a whole lifetime of experience. Languages are embedded in culture and personal experience, such that semantics require that context even if syntactics do not. Vaguer languages such as Japanese, can convey much subtler meanings than English or German. There is possibly a way to maximise a weighted sum of the creativity which ambiguity allows and the accuracy which precision allows. Being a mathematician or a physicist like myself we are used to treating mathematics as the ultimate tool in understanding nature.

Philosophy of Mathematics - Bibliography - PhilPapers

We develop mathematical formalisms that describe nature and then marvel at how well mathematics is suited to describe our subject of interest. But is mathematics really so universal as we make it out to be, or do we just see it in that light because we are so used to it? The foundations of mathematics evolved in our ancestors for the need to count things in order to have to survive.

Mathematics was used for centuries by the Babylonians, Egyptians, Greeks and other highly developed cultures. But it took the genius of a Newton or Leibniz to develop differential calculus and to be able to properly describe and object's motion when a variable force acts upon it. This formalism uses infinities and infinitely small intervals and is quite a long way from the simple act of counting.

So my answer is: The world can be understood mathematically, only because we develop the mathematical tools that allow us to understand it. Constructions, like differential calculus, do not follow inevitably from the basic principles of mathematics, but only from our need to describe nature. But this reverses the dependency. The universe is not inherently mathematical, but mathematics is constructed to be universal.


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Yes, this seems to me to be very good and valid point. We identify a subset of real-world experiences all of which are well-described by mathematics and then step back and marvel at the "universality" of mathematics. I'm reminded of this. Nevertheless, there remains the question of exactly why we have access to a "language" to take George's point below which we can perhaps abstract from a subset of the world, play with in our minds, and then accurately re-apply to a subset of the world.

We take this for granted, but, as I am trying to establish, this fact is stunning and perhaps non-obvious. A mathematics fan myself and a self proclaimed philosopher : , though everyone is a philosopher in their own rights , I do believe in the universality of mathematics but feel that mathematics is just an aspect of reality and not an all pervading foundation. Mathematics can be seen as a language of describing certain kinds of correlations between objects, events etc.

So whether mathematics is all pervading depends on whether i Everything in this world is correlated ii Can mathematics describe all correlations in this world. Irrespective of whether i is true or not, I think its not too difficult to construct examples to show ii is not true. As many would already have thought of, feelings like love, etc cannot be explained by mathematics. Reminds me of this valentine's day quote by H. Mencken: Love is the triumph of imagination over intelligence. I can say this to extend Holger's comment: Most of the things or phenomena around us can be measured if not objectively, then perhaps subjectively , and measurement is very much mathematical.

So one may feel that everything is mathematical. But in reality, measurement or correlation between entities is just one aspect of their existence. So its wrong to say mathematics can describe or is the basis of everything. I did enjoy aspects of this article..

A nice point made by Anonymous and PhilWilson. It is true that relations of the heart are not the same as relations of numbers.

Stewart Shapiro

People not believing this no mathematical equation can predict what gift gift your valentine will want on this Valentine's Day: Having said that I think there is no end to the argument whether emotions like love are matters of the heart or produced mathematically controlled events in the brain. And now on a lighter note, for all those who believe that mathematics can describe everything including love, two proofs:.

Firstly, some gems of quotes from this page: mathematical love quotes for valentines day :. Now this is one great use of applied mathematics, right? Caution: Use the above quotes only if your valentine is a geek :. Also do read this interesting article on valentine programming apparently people are researching how mathematics can explain lost love and what not.

New Waves in Philosophy of Mathematics

Applied mathematics is a whole ocean out there. I appreciate the appeal of your question. I was a philosopher before I began seriously studying mathematical physics, and even though I had already considered the basic questions, I found myself struck anew by the regularity with which nature has been found to conform to the most abstract imaginative leaps of rare minds. Nevertheless I believe that the question is more to be dissolved than solved. It is akin to the question of how language can "latch onto reality" the answer to which lies in understanding our own behavior.

Some of the questions you ask e. An amoeba responding to being poked is already "comprehending" a tiny piece of the world in a sense, and higher animals display a finely-tuned set of useful, reality-conforming behaviors requiring discrimination and goal-pursuit. Their mental constructs "conform to reality" because and to the extent that they worked in the past and others didn't.

And of course we know that past results are no guarantee…. We can contrast understanding here with ignorance there, or complete understanding with partial understanding; but we have no concept of a world without any understanding at all. So this is why I say that your question is at bottom a pseudo-question which may nevertheless be very educational and fruitful along the way to being dissolved.

Aeon for Friends

I don't think your 4 positions are exhaustive. I'd say you're missing a position of Realism, which says that mathematical structures inhere in nature, without requiring any "independent realm" to give them "reality". It is nature that gives them reality. For we cannot conceive of a nature without mathematical structure--without unity and multiplicity, spatial orientations, boundaries, sets….


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And by the way I'd like to add this open-ended list of mathematical categories to your account of the natural genesis of mathematics out of counting. Not just counting, but geometrical intuition and discrimination of natural sets seem to me to be equally grounded in an evolutionary account. You say that a theistic picture is tidier than the others, but are afraid of embracing religious baggage.

I do think that a Spinozan patheism is a viable solution, as long as you realize that it is really nothing more than a counterweight to the kind of materialism that denies any structure or intelligibility to the world independently of humans. To call the world "divine" in this sense is just to acknowledge that before intelligent beings, the world already had intelligible structure, and contained the potentiality for the causal generation of intelligence.

Mysteries of the Mathematical Universe

I am currently attending a workshop at the American Institute of Mathematics and am very busy indeed, so I am afraid that your answer will not get the full and in-depth response which it merits. My apologies. I think that your raise some excellent points. While I am very receptive to the evolutionary approach - and while I think that it can be extended to provide a definition of mathematical beauty in terms of maximum fecundity - it leaves me thinking "and yet, and yet.

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