Mathematical entities - do they exist ouside the human mind?

simu

Do mathematical entities (numbers etc) exist independently of the human mind? Are mathematical concepts discovered or are they invented by the mind as a tool to solve problems?


There's a pretty good page introducing these topics here:http://en.wikipedia.org/wiki/Philosophy_of_mathematics


I would be most inclined to believe the embodied mind theories (explained on said page). The idea that language and ideas are a product of the human mind and would, perhaps be incomprehensible to alien beings does not seem very surprising - are mathematics really that different?


On the other hand, maths can be used to describe the physical world and areas of mathematics that were thought to be completely abstract have been found to have practical uses when new scientific discoveries were made. From this arises the indispensibility argument - to quote from the wikipedia article :" mathematics is indispensable to all empirical sciences, and if we want to believe in the reality of the phenomena described by the sciences, we ought also believe in the reality of those entities required for this description."


I'm not all that convinced by this statement - it smacks of "we must believe this because other wise, the consequences are horrifying". Is it not possible that the phenomena described by maths are real (i.e. existing independently of the human mind) even if the maths are just a tool invented by the human mind to describe them? Is it any different from a dog being real in itself whereas the word "dog" is an invention of the human mind to allow us to think and communicate about the phenomena of a dog. And similarly, the concept of a "furry green alien", a concept produced by the human mind, could someday become useful to describe the inhabitants of a distant planet even tough it only refers to an imaginary idea at present!


I'm curious as to what people (whether they be mathematicians or not) think about this!

the raven

that sounds like its kinda a good few rungs up the ladder from Husserls initial phenomenology theory? (damn interesting stuff... can't wait for next year - in 2nd Arts we get to study the phenomenology of Cezane's works!)

DadaKopf

Yeah, Husserl was big into this. However, I was never convinced by Husserl entirely, but I'm convinced of numbers and I'm convinced of the phenomenology people built on his observations.

I always just kind of figured there were different orders of mathematics, and logic, I guess. Numbers are sort of operators that exist in the nexus between ideal and material realms, they are more specific than signs and so can never be separated from their references in the same way most other signs can.

I'm not an essentialist, so I don't think, like Husserl thought, that meaning is something essential to an object that can only ever be intuited. I think things like that are socially constructed.

But it always seemed to me a necessary ambiguity in numbers that they are abstract symbols of mathematical concepts and signs for something are unchangeable. Sure, the the visual signs can change, but the twoness of two is something that's unchangeable no matter what critical models you employ. If that's a return to Husserlian phenomenological method, then so be it.

For everything else, I think things get so complex that we can only see things as social constructs - higher mathematics, in fact, any theory is just subjective experience codified as a model.

keu

I have to admit I'm guilty of failing maths (lack of interest) but I honoured in art, and through this avenue I discovered sacred geometry. You mentioned cezanne, I recall learning about the golden ratio, can't remember the artist who first applied this formula, (where the work is divided up into "divine portions")

but I understand it is derived from the study of natural geometry (patterns designs structures ie: honeycombs)

will have to leave it there because I don't remember enough about it to discuss it in any depth....
http://www.intent.com/sg/

DadaKopf

It's the irony of modernist art that people like Piet Mondrian and Kasimir Malevich, or architects like Le Corbusier, through attempting to discover the universal laws of aesthetics contributed to the failure of any attempt to objectify human experience through mathematical/geometric rules. They failed, so we got postmodernism.

I'm currently big into photography (I like the line it draws between objectivity and representation) and I love plenty of photos that don't obey those classical rules, and I love many photos that absolutely do.

You're right though, compositions that follow the classical rules of aesthetics (golden section, rule of thirds etc.) are usually winners. But also very often not.

So, maybe I should go back and reconsider everything I said in my previous post! Jeez, I have to get back to reading my aesthetics books!

DadaKopf

Actually, I haven't changed my mind.

But, just to extend the conversation a little bit, I just read this in a book:

Let me begin with a strong, simplistic, and usefully wrong definition of objectivity: a given perception, recognition, or understanding can be called 'objective' if its content is wholly or largely determined by its object - so that a range of human subjects, differently placed, with different personalities and different, even conflicting, interests, would agree on the same content so long as they attended to the same object ... the object imposes itself ... [t]he subject is passive and undiscriminating, a promiscuous consumer of available data.

For reasons philosophers have long understood, this cannot be right. Human beings are active subjects.


Michael Waltzer (1993), "Objectivity and Social Meaning"; from The Quality of Life, edited by Martha C. Nussbaum and Amartya Sen.

SkepticOne

I would tend to go with the "embodied mind" theory too. It means that there is no need to invent a separate independent realm of mathematical truths while at the same time explaining why many agreed mathematical results don't appear to be empirically verifiable.


On the Social Construct theory:

This theory sees mathematics primarily as a social construct, as a product of culture, subject to correction and change. Like the other sciences, mathematics is viewed as an empirical endeavor whose results are constantly compared to reality and may be discarded if they don't agree with observation or prove pointless. The direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it.

Interesting the phrase "... and may be discarded if they don't agree with observation or prove pointless." This would be assuming that something is being observed? What would this be?

But more generally, is the social construct idea saying anything about mathematics? Is it saying that, for example, "2+2=4" is simply a reflection of Western cultural biases (for example). Some other culture or social group might well have it different? If not, then what is it trying to say.

While no doubt there are influences from society, a problem for this theory is how do you separate these out from what might possibly be in-built tendencies in the human mind arising out of evolutionary forces on the brain.

simu

Interesting the phrase "... and may be discarded if they don't agree with observation or prove pointless." This would be assuming that something is being observed? What would this be?

I take it they're referring to maths being used for scientific purposes - if a particluar "piece" of maths isn't useful for the task at hand, you look for another way of getting the job done. So, what's being observed isn't maths but rather, maths is being used to study something else that is being observed.

But more generally, is the social construct idea saying anything about mathematics? Is it saying that, for example, "2+2=4" is simply a reflection of Western cultural biases (for example). Some other culture or social group might well have it different? If not, then what is it trying to say.

Well, to quote the article : "The direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it." So, maybe if the Mayans or some other group had developed mathematics and become a dominant culture in the world or if women had had a greater role in maths in the past, it would have taken a different path. As for 2+2=4 being an example of Western cultural bias - well, I don't think any human tribe has ever been found that didn't have basic counting. Even if such a tribe were found, it seems likely that it would be possible to explain the idea to them. It is thought that even some animals have very basic counting ability:

A theory held by some is that humans and other animals share a basic neural system called an "accumulator" that can clearly distinguish numbers of objects less than three or four but that cannot reliably discriminate between bigger numbers. This accumulator is active in animals and, perhaps, in human infants, the theory contends. Higher-order number abilities require the collaboration of other, more highly developed brain systems found only in humans.

from: http://www.apa.org/monitor/apr99/math.html

SkepticOne

Originally posted by simu
I take it they're referring to maths being used for scientific purposes - if a particluar "piece" of maths isn't useful for the task at hand, you look for another way of getting the job done. So, what's being observed isn't maths but rather, maths is being used to study something else that is being observed.

This is a fair assumption and I would agree with what you say. However I'm thinking more of mathematics generally rather than applied maths. There are plenty of results in mathematics with no known applications, applications that are unknown to the mathematicians making the discoveries or applications that are discovered years after the initial mathematical discovery (or invention depending on your viewpoint).

My points about the social construct theory may be unfair, however, if the theory is not about the underlying nature of mathematical objects but rather about how mathematics is put into practice in solving real world practice.

Well, to quote the article : "The direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it." So, maybe if the Mayans or some other group had developed mathematics and become a dominant culture in the world or if women had had a greater role in maths in the past, it would have taken a different path. As for 2+2=4 being an example of Western cultural bias - well, I don't think any human tribe has ever been found that didn't have basic counting. Even if such a tribe were found, it seems likely that it would be possible to explain the idea to them. It is thought that even some animals have very basic counting ability.

The 2+2=4 might be a bad example. While I think most people would agree that mathematics moves in directions influenced by the needs of society (e.g. a country wanting to develop neuclear weapons may well fund mathematics required for the understanding of physics in that area, this does not say anything about the nature or reality of the mathematical entities themselves. Archaeology is influenced by political requirements. I believe the Nazis funded archaeological activity to justify expansion and I'm sure there are more examples. However this does not say anything about the independent existance of the objects under study in archaeology. The objects are either exist or they don't.

simu

While I think most people would agree that mathematics moves in directions influenced by the needs of society (e.g. a country wanting to develop neuclear weapons may well fund mathematics required for the understanding of physics in that area, this does not say anything about the nature or reality of the mathematical entities themselves.

Yeah, it's the nature of mathematical entities themselves I'm most interested in.

Anyone who has studied maths at uni, is this issue ever raised in lectures?

The reason I strated this thread was that I was reading a book about the Riemann Hypothesis and there was a chapter in the end of the book about whether maths are discovered or invented. The writer interviewed different mathematicians on this and many of them believed that they were definitely discovering or else they preferred not to think about such things at all.

Fysh

On first impressions, I'm not entirely able to decide what view I agree with most...I'd probably go for something like the embodied mind theory, since it seems likely that our brain would adapt in such a way as to be able to handle those mathematical ideas which might prove useful (this notion does fall apart somewhat when you start talking about advanced maths, since there's no easily discernible way in which an understanding of, say, second order differential equations will have a direct influence on the probability of survival of an individual, but I digress).

I'm surprised nobody's mentioned the Langlands Project or the Omega number yet. The Langlands project was discussed briefly at the end of "Fermat's Last Theorem" - basically a project to fund research aimed at discovering links between different branches of mathematics, hoping that being able to translate mathematical problems between different formulations might reveal new ways of solving previously unsolvable problems.

The Omega number was something I read about in New Scientist a couple of years ago - I'll need to look around for some online resource about it, but basically it was a line of research which investigated convergences of certain types of sets. I honestly can't remember the detail but basically the research was expected to conclude that all mathematics was random, and that there was no overall pattern or superstructure.

The reason I mention them is that the success of one or the other would surely offer some insight into the discussion? As in, if the Langlands projects repeatedly finds links between different branches of mathematics, this would suggest (Although not prove conclusively, of course) that there is some underlying structure - an idea compatible with a dimension in which mathematical entities have some physical existence.

I'll look and see if I can find more information on the Omega number and post a link...

Fysh

Update - I found the article I was talking about:

"The discovery of Omega has exposed gaping holes in mathematics, making research in the field look like playing a lottery, and it has demolished hopes of a theory of everything. Who knows what the Super-Omegas are capable of? "This," Chaitin warns, "is just the beginning.""

The full text is available here.

This guy seems to go for the maths as discovery view, although I would argue that if (as he seems to have proved) maths is basically random, then surely you can consider the act of finding a mathematical tool or process that works and is useful as being an "invention" (in the way that you can say that Newton "invented" calculus as a means to solve an otherwise intractable problem)?

Sir Digby Chicken Caesar

regarding the article about animals and numbers... i'm reading a book atm "georges ifrah - universal history of numbers", about how counting and number systems, and the numbers we use today came to be over the course of human evolution. it's a pretty interesting read and somebdy that is interested in this thread might do well to have a look for it ('s on amazon i think).

(actually, the first bit in the book is about animals counting )

Uthur

Funnily enough, animals have a 'number sense' which can function
better than the equivalent in human beings in some species. Chimps
can tell if they are looking at between 1 and 8 objects and press the right
number on a keypad very rapidly. I guess in a way animals can count!
I'm not sure exactly how this relates to the argument, but if numbers
at least exist only in the mind, they don't just exist in the human mind.

earwicker

Mathematics is a language, and I don't see why it shouldn't be subject to the same vagaries as any other sign-system: meaning, interpretation, translation, slippage. It's just that some members of the interpretative community that makes mathematics by controlling the meaning of the language are open to experiencing the problems associated with sign systems.

It seems that much of the newer mathematics are understandable in terms of semiological problems.

ecksor

Hi,

I'd appreciate if you could elaborate upon this:

It's just that some members of the interpretative community that makes mathematics by controlling the meaning of the language are open to experiencing the problems associated with sign systems.

and this:

It seems that much of the newer mathematics are understandable in terms of semiological problems.

Thanks.

PhilH

Mathematics is a language

I would suggest that this statement is at the heart of the thread's original question. We have a system for talking about mathematics (the written symbols for numbers, operations, etc.) which are different in different cultures. So different people might use a different symbols to represent 'two', 'plus', 'equals' and 'four'. Hypothetically, over time, the meaning of some symbol might change (e.g. from 'equivalent' to 'exactly equal').
However this is independant of whether or not the concepts being described are in some was independant from the human mind. To go right back to the start, the word 'dog' might change, and is 'an invention of the human mind', but the concept 'dog' presumably exists independantly.

So, I would suggest that mathematics is not a language, but we have languages for describing mathematical concepts.

earwicker

Brief hit and run response: Brian Rotman has done a lot of work on this:

"Toward a Semiotics of Mathematics" Semiotica, 72 (1), 1988:1-35.

"The Technology of Mathematical Persuasion" in Inscribing Science, (Edited Tim Lenoir), Stanford University Press, 1999

"Thinking Dia-Grams: Mathematics, Writing and Virtual Reality" South Atlantic Quarterly, 94:2, (1995), 389-415

"The Truth About Counting" The Sciences,Nov/Dec, (1997) 34-39

He also has a new book coming from Standford Uni Press called Mathematics as Sign: Writing/Imagining/Counting