Philosophy discussion

There has always been something bugging me. I've always found mathematics a fascinating subject but I could never put my finger on what exactly mathematics is. Perhaps, you might think, I would be better off asking mathematicians this question, perhaps, I don't know, however I would still be interested to hear what the rest of you have as an answer or, in general, have to say. Maybe there isn't some one sweeping definition.

Cool topic. I'm headed back to college having been laid off from Verizon in Nov. I work in networking but have no degree so I'm going back as an engineering major. I've always been very good at logic and saw math as a time consuming language to learn, which I did not have the patience or time to commit to.

Now that I'm going back to school I've been brushing up on my math, using a free online math tutorial called Khan academy. In the last 2-3 weeks, since I've had plenty of time to invest in mathematics, what I've noticed is that math is in fact a language which describes the nature of the universe. I'm not certain there is an absolute truth to mathematics because it is based on the human categorization of quantities, but I am very certain it is the best description for reality we have.

Our native languages do an excellent job of describing cultural affairs and literature does the job of describing emotions. Mathematics seems to play a similar role telling us the story of quantities and causalities. I found my new interest in math to be very stimulating, though I know I could never have found the passion for it in my younger years. Traveling with a band was far more captivating.

I very much agree on the idea that mathematics is a language, a kind of tool to organise data and to help solve problems. However not all of what is studied in maths can be applied. What I am interested in is more what is it about maths that makes it maths (if that makes sense to anyone). What makes science science is that it, as a subject, requires making observations and from those observations constructing theories. Another question that somewhat, I suppose, goes in combination with the question of this subject is what distinguishes maths from logic? Or are they in anyway different?

Maths, as most people think of it, is logic applied to the concept of quantity. That doesn't cover everything, but if you add "..and logic(s) too" then you're just about there.

I agree thats actually a pretty good point Bart. To add to that I somewhat like Marcus du Sautoy's definition that mathematics is the searching for patterns in numbers.
Awesome name by the way Bart!!

Another important question is whether math is just a "language" as Mrtfalls puts it, or an expression of something that exists in its own right. Put in a different way: would 2+2=4 even if there were no sentient beings to appreciate that fact?

There's definitely something called "mathematical literacy," just as there is with alphabetic literacy, so that implies some sort of language.

But I'm not sure it would exist at all without a knowing observer to express it. As Brian said, it's based on human categorization. Also, I've come to wonder whether we can invent a mathematics to describe just about anything humans wish to. Does that sound right?

Rebecca wrote: "Another important question is whether math is just a "language" as Mrtfalls puts it, or an expression of something that exists in its own right. Put in a different way: would 2+2=4 even if there we..."

I see what you're leading onto Rebecca, this is a problem A.N. Whitehead and Bertrand Russell had there question was "is mathematics a mental concept or construct designed by humans or is it something out there, something to be discovered.
Well hopefully I've understood what you were leading onto.

Patrice wrote: "And isn't music based, in part, on math?"

Describing music and understanding why some note and chord structures sound a certain way is all based on mathematical principles, but one could hardly say Pink Floyd's Darkside of the Moon was based on mathematical principles, rather it used mathematics to describe its own story, that being the music of the album.

This leads directly to Rebecca's question of whether or not math exists as something mind independent. I do not believe math is mind independent because it is mind that makes math out of worldly interactions. To borrow an idea from Kant, where he held the notion that time and space were innate faculties for understanding the nature of reality, I think mathematics is an innate quality for being or existing temporally.

I'll break it down this way; if there were aliens who came to earth after humans completed the annihilation of their species from war or disease or something, would the aliens be able to understand our math or better yet, would math exist for them, or even at all? I want to say no. No because it is very possible that the mechanical system they used to build a spaceship and navigate to our Earth was based on something which was intrinsic to their perception of reality. Maybe they do not understand reality in a dimension equivalent to ours and mathematics is not exactly a fundamental principle for generating the proper mechanics for space travel. For instance do honeybees or ants need mathematics to complete their objectives? Maybe we can see a mathematical pattern in what they are constructing and trying to achieve but that is due to or own necessity of seeing the world mathematically.

Again back to Kant. Kant thought our interface with the world resided on an innate principle of Euclidean geometry, but he was wrong because we can understand the world in non-Euclidean terms as well. The idea is still powerful, that humans have innate abilities to understand reality via some pertinent faculty. Mathematics seems to be one of those, just imagine yourself existing without it. I don't think you can, and that is why I think mathematics is necessitated by human existence.

If human understanding of mathematics is synonymous with the aliens understanding of reality through their particular mechanics that doesn't make mathematics universal, rather it makes it incomplete because it would become a part of a greater unperceived totality.

Oh, and Tyler I think humans can invent a mathematics to describe just about anything. I think many have tried as the aforementioned Russell and Whitehead project would show. I think the trouble starts when the problem of language permeates a discussion like that. I mean just imagine someone trying to translate a mathematical description of certain phenomena, and compare that to the popular notion of statistics when applied to something like disease. I mean, how many people don't treat their children for curable disease because they do not understand how averages work. So laymen would be lost and mathematicians would be one sided. It might work out for the occasional polymath. : )

Patrice wrote: "Not sure I understand Tyler.

When I was learning to be an elementary school teacher we were taught that there were three parts to teaching math. There was the real, concrete, physical reality. C..."
I personally think math would become the universal language with aliens. Do you find that children have much more difficulty with math problems that are in written form rather than just numbers? I'm sure you do, as I did when tudoring in junior high math. Language can definitely make math more difficult to understand.

Music is partially mathematical but it would be difficult to use music as universal language. We would have difficulty in using it ourselves as very few people have perfect pitch. We could all learn to read music but it wouldn't sound the same.

Einstein's brain had some differences . It is my understanding that the areas of the brain which many of us use for spatial perception were larger in his brain. On the other hand, we now know that people may use different parts of the brain for the same acitivities. Like usual, the more we know , the more complex the question and answer (if there is one) .

(I've seem to have lost another comment Language makes math more confusing. I think most people find a math question in written form is much more difficult to answer than if the question is in just numbers. I believe it would be the most universal language and the closest thing we have to being absolute. But we also know that macro concepts like Eisnteins work but cannot be completely confirmed at the micro level.The more math you know, the less absolute it seems to become. In that respect it is not different from other fields, you just have to reach more difficult mathematical concepts before you realize there is probably no absolute. We don't know if matter will not always act in the same way,

The ability to learn languages at a young age, the bility to play music by ear,the ability to draw in perfect perspective with no training, the ability to math at a very high level without training. Are these inherited skills, some more common that others, just inherited or is there also something else as well at work.

We need separate familiar equations and our contemporary human practice of mathematics from mathematics as the abstract family of models of quantity, space, relationships. Whether or not Brian's aliens and bees formalize math the same ways we do, if they are reasoning about distance, angles, velocity, and other relationships of quantity and space, then they are constructing a system of mathematics. Theirs might be inconsistent or lack features we find elementary and they may entirely lack interest in proving theorems or discovering new properties, but it's mathematics nonetheless.

Here's a blog post and discussion readers of this thread might find interesting: http://scienceblogs.com/goodmath/2009...

Couldn't it be mathematics to us, but some other mechanical description of reality to the other being? A system based on sense of smell for instance. We could say that the alien's moves reflect a logic where; if it smells stronger here then there, walk three paces and smell again. It could be more complex and add the following instruction, if it smells stronger here again, then walk 5 paces and smell again.

We can mathematize this navigation of the alien as a function of the scent. We describe the whole event in mathematical terms, if x then y, or perhaps as; 3 steps equals the function of scent, or some other equation, but it is still mind dependent since the ant or alien does not perform mathematical action, they follow only a scent of smell and intuit there next move based on previous action. If they decide to describe the action mathematically they could, but that does not mean the equation existed always in a transcendental realm, but rather the mathematical user needs to be in order to do the calculation.

That's why we need to separate mathematics-as-practiced from mathematics-as-abstract-reasoning (or however we want to classify it). What you described through the smells is a finite state machine. It's math*, just not the plane geometry we might think the alien should be using to navigate. The equation "exists" in the neurological structure of the alien (or whatever one calls their thinking parts).

(* EDIT: That probably should be "It relies on mathematical relationships..." or something akin to that.)

Galileo said something like (forgive my horrid translation skills, Latin was never my forte!) "Mathematics is the language with which God wrote the universe." I used to have a copy of Elements by Euclid, and from what I got out of it was that math is not exactly a science, but sort of a set of rules for nature. Best non-philosophical definition I've found? Compact OED online says "the branch of science concerned with number, quantity, and space, either as abstract ideas (pure mathematics) or as applied to physics, engineering, and other subjects (applied mathematics)." So, mathematics is really one field that is applied two ways, either to hard sciences such as physics, chemistry, etc or to more abstract things. Two sheep is two sheep, sure, but numbers are really abstract.

So, perhaps the best way of defining mathematics is that it is the abstract science of describing the logic of the natural world...

P.S. I've always hated math, so I would never call myself anything more than a layman in this field...

It is unfortunate that being in a particular field may give one greater insight into the intricacies while never offering any greater understanding of the whole. Thus as mathematics was my first undergraduate degree, one thinks that while I might have a greater understanding of what it is, alas, I doubt that this is so. I had an affinity to the branch known as Analysis. We would make fun of the Algebraists and they, in turn, of us. The only thing upon which both of our groups agreed was that the Topologists were all crazy and delusional. It was especially revealing when I discovered that engineers made fun of ALL the mathematics: all they wanted was a number they could use! That was as cohesive as the world of mathematics ever became for me.

Indeed I believe that all legitimate mathematics describes something of the real world. However I do such on mere faith, albeit heuristically. It is unfortunate that we do so much of it which does not yet have a practical application, but we so hope. As Whitehead and Russell tried to derive a concept of Mathematics from Principia Mathematica, a truly marvelous book, it is also amazing that Kurt Gödel came along to dash their efforts. I enjoy reading the discoveries of the Polish mathematicians to whom we owe homage for the calculator’s humble beginnings) as well as the logic of Frege, Wittgenstein, Moore and Peirce. The mathematics of the early 20th century is surprisingly fascinating.

Still if nothing else, I believe that mathematics mirrors the logic of our human minds.. Hence it may not be so far fetched, unless one believes that mathematics is rendered by some spontaneous generation (which places you somewhere beyond the Topologists in my estimation,) that whatever mathematics encompasses by definition, its source is the mind of our creator.

Mathematics is the formal language that emanates from Plato's world of ideal forms.

It is the link between the objective physical world and the world of the conscious awareness of self.

One plus one only approximately equals two!!!

Quantum theory teaches us that exact equalities do not exist except in a probabilistic sense.

Oneness or unity by definition is unique and cannot be added to another to form a pair without redefining the class of object created.

I never liked word problems in junior college, and that was a language that was very confusing. Mathematics to me is applied logic as far as algebra there is a formula, once you have the formula you know what the answer is, or what is the next step to getting the answer.

I don't know if Einstein said that, but Mathematics is sure the language of the universe.

I'm not sure I'm repeating this right, but years ago in college, a professor told the class that everything in this universe can be reduced to a mathematical equation...everything. He said it is a strange phenomenon that occurs in a world that many people deem having resulted from chaos.

Hi Sandysconnected --

I recently read a book that the professor might agree with, What Is Philosophy? The authors look at science, including mathematics, as a way to "transect chaos."

According to them, human minds have three ways to transect chaos, to make sense of the universe; the distinguishing feature of the scientific approach is a coordinate system of some kind or other, which yields functions, which leads to propositions about what's out there.

A coordinate system and functions are of course basic to mathematics. The authors of this book would agree, I think, that from a mathematical way of thinking, the universe might very well be reducible the way your professor says.

Sandysconnected wrote: "I'm not sure I'm repeating this right, but years ago in college, a professor told the class that everything in this universe can be reduced to a mathematical equation...everything. He said it is a ..."

You can't reduce Proust or the lessons of history an equation.

Tyler wrote: "Hi Sandysconnected --

I recently read a book that the professor might agree with, What Is Philosophy? The authors look at science, including mathematics, as a way to "transect chao..."

The operative words here are "from a mathematical way of thinking." But what mathematics can quantify is only a small portion of what we know.

Mathematics is just one of the many means of communication that humans have developed/evolved to help describe reality as perceived by the individual.
It has proved very useful in describing those aspects of reality that have recurring action - that can be modeled in a spacetime framework. It application to more complex areas such as biology or weather systems has been limited as these are non-reducible systems. Stephen Wolfram has some radical proposals to chart new directions for mathematics in such complex arenas.
There are many other languages other than mathematics that help us communicate our otherwise solipsistic existence to our fellow human beings - music, art, literature, and the host of 'soft' sciences. Oh and of course philosophy.

This goes nowhere in telling is what mathematics is, or even what it isn't, Desgreene. Sort of like your posts in "Philosophy is Dead."

Bart wrote: "That's why we need to separate mathematics-as-practiced from mathematics-as-abstract-reasoning (or however we want to classify it). What you described through the smells is a finite state machine. ..."
I found Math a constant struggle growing up. I respect Pythagoras, Einstein, and their brethren for their own struggles with the concept. It's as easy as 1+1=2 but as difficult as the theory of relativity.

Hi John --

The operative words here are "from a mathematical way of thinking." But what mathematics can quantify is only a small portion of what we know.

That's correct as far as I understand the authors. They propose two other planes, together with the scientific one, in order to achieve a comprehensive view of the world.

Deleuze and Guattari are usually a little radical for my tastes. I haven't read that particular text, but "A Thousand Plateaus" and "Anti-Oedipus" were very odd going. Before I clicked on the link, I thought you were referring to an elementary text. Only now did I see who wrote it.

Hi John --

This book is quite odd, too, and filled with abstruse ideas. I was surprised that what the professor said about mathematics squared so closely with the authors' notions concerning science.

Which professor are you talking about?

Sandysconnected's professor, in message 27

From what I know about Deleuze and Guattari, they hardly would have agreed that all knowledge would have been reducible to equations - if that's what you're saying, Tyler. One of them was a psychoanalyst. I don't even know how you'd reduce rhizomatic theory or transference into a mathematical equation.

If I remember this correctly, the scientific plane that transects chaos reduces to a series of functions and coordinates. It sounds materialist to me, but they propose other planes that also transect chaos and account for the human element.

I didn't find anything wrong with this approach, but it is quite different from my own understanding. For instance, concepts fall into one plane and propositions into another, and it's not clear how that makes for better philosophy.

It's true, too, that they gave few examples (that I can remember) showing how these planes work together to account for something like a theory.

I see a lot of people here claim what I once claimed, that math is a language. I think this is the result of viewing math too narrowly, though, as just a collection of words and symbols that get used in some linguistically permissible ways and some linguistically impermissible ways (An example of violating math grammar is easy: 1+1 → {R/S}. All of these symbols are employed in math, but being employed in this way cannot even be said to be true or false, but simply meaningless.). Instead, I think we should take stronger heed of how these symbols and words get used in social practice.

There are two simple and inaccurate outlines of how math gets done—and how it really gets done is a complicated mix of the two. But it's worth describing them, to make my point:

1) A mathematician works with some kinds of natural mathematical structures for a long time (integers, Euclidean geometry, etc.) and eventually muses about changing the structure either by adding some extra mathematical objects (rational numbers, for instance) or removing some axioms (Euclid's Fifth Postulate comes to mind), or in some other way merely playing with pre-existing structures in some way that comes up with a new pet structure to study. This is then understood well by a number of people, who realize that it's very apt for describing some practically useful, physical system, and it gets implemented as a major tool in some scientific discipline.

2) A scientific discipline has some challenges in understanding or calculating some feature of its study. It then seeks out mathematicians to either create a useful mathematical structure that describes their studies, or to develop theorems concerning existing mathematical structures which allow them to be used more efficiently. An example of the latter is that, by realizing that decay of radioactive substances is proportional to the mass of the radioactive substance, we find out that we want a mathematical way to describe relationships between quantities, their rates of change, rates of acceleration, and so on→thus was born the study of differential equations, developing theorems like the existence and uniqueness theorems that allow us to stop looking for extra solutions once one solution has been found. An example of the former might be the graph theory that's been applied to computer sciences, or the development of category theory.

So now I'll deliver my point. Rather than mathematics being merely a language, I believe math is actually a codification of precise rational thinking. We do not merely have words and symbols for grammatical sentences, but built into the system itself is the standard of what is true and false. 5 + x = 10 ⇒x = 5 is just as grammatical as 5 + x = 10 ⇒ x = 0, but the former is required by the axioms and inferences chosen to be true, while the latter is not. No language, except in a very pliant idea of what a language is, is designed with permissible inferences built into it.

Instead, mathematics is a pre-investigated set of rational deductions. Need to find out how much an item once cost, based on knowing how much it currently costs and the percent change? Here, have some algebra! We've already figured out exactly how to rationally understand all such scenarios and make a rational deduction about the previous cost. Sure, you could figure it out on your own. Say it costs $50 and it's the result of a 50% reduction in price. That's not too hard to figure out: What dollar amount, when reduced by %50 is equal to $50? Well, 100 springs to mind. That's not even really mathematical reasoning, but just sort of quick, natural insight. But if you have a harder problem, mathematicians have realized that you can always analyze the situation as: New price = Old price + (Percent change)(Old price). Just feed the specific number of your particular problem into this formula, run the algebra mindlessly, and you get the answer.

Even more advanced mathematics works this way, just in more complicated ways. Trying to find the area of a shape that's described by the function f(x), and want to know whether the integral ∫f(x)dx exists before you go trying it? Well, show that it's continuous and you'll have what you want. You know that, because some mathematician long ago went through the reasoning pattern to show that all continuous functions are integrable—so now you don't have to. He, in a sense, pre-digested the reasoning for you and now you can skip it so long as you trust him. How did he do it? Because the mathematical structure of the field of real numbers was designed to "describe" or reason about physical scenarios (like the volume you're trying to find). Any form of reasoning that applies to the real numbers, will roughly apply to what is roughly like the real numbers. And at least when dealing with middle-sized dry goods, the physical world is like the real numbers. Space has three dimensions, each line segment seems infinitely divisible (at least to many many decimal places of approximation, so we don't really have to worry about that), and everything seems like it should be algebraically complete.

Hopefully I've communicated my basic claim well: Mathematics is not itself logic (I haven't really addressed this much in my post, but it should be obvious that I don't think it is—logic concerns reasoning about ANYTHING, whereas mathematical subjects are concerned with reasoning about whatever particular subject their structures are designed to reason about), and math is not merely a language. It is a set of techniques of reason and conclusions about subjects that have some given structures. I know it's arrogant to say it, but I like using the slogan, "We do the thinking so you don't have to!"

I also think that, for any proposal about the nature of mathematics as a whole, we shouldn't always appeal to physics and counting physical objects as our examples. We should also look at set-theory and topology, which in themselves really have no notion of distance at all, so it's hard to see them as specifically describing a physical system. What are the physical systems that probability and game theory describe? The two answers seem to be, 1) the multiverse of all possibilities or 2) it's not a physical system but a property of physical systems. I think both of these won't work. 1) won't work because if that's so, then why should we care about the multiverse rather than just focusing on our own universe? But if we study the multiverse with agnosticism about which universe is our own, then when we apply the lessons to our own lives, we're no longer thinking of it as the multiverse and it seems like what we really studied in the first place was rational belief. 2) won't work because it depends on a frequentist interpretation of probability, and that's seen enough criticism that I don't feel like rehashing it.

A similar problem is confronted with mathematical logic. That doesn't seem to be analyzing a physical state, but studies the results of attempts at formalizing rationality.

Hi Adam --

Thanks for the detailed and informative look at the nature of mathematics. Here are some comments:


--Rather than mathematics being merely a language, I believe math is actually a codification of precise rational thinking.

I've heard mathematics described in terms of language quite often, but I've always felt ambivalent about it, wondering if it were an inexact analogy. This way of looking at mathematics may stem mainly from the linguistic turn in philosophy: When you have a hammer, everything around you starts looking like a nail.



--Mathematics is not itself logic (I haven't really addressed this much in my post, but it should be obvious that I don't think it is—logic concerns reasoning about ANYTHING, whereas mathematical subjects are concerned with reasoning about whatever particular subject their structures are designed to reason about)...

This is another point of confusion in philosophy. I've heard it said so often that logic is the sole preserve of mathematics, yet I've never heard an explanation as to how existential statements must have a mathematical basis to count as logical, or how that basis could be established in the first place.



--...for any proposal about the nature of mathematics as a whole, we shouldn't always appeal to physics and counting physical objects as our examples.

The "Philosophy is Dead" thread demonstrates, in the words of Stephen Hawking no less, how physics has come to be seen as that to which all other subjects are reducible. People disagreeing with this often seem to be fighting an uphill battle. My take on it is that even if you could reduce a field of inquiry to something expressible by physics, it would do little good. It would be like talking about biology in terms of chemistry. Sure, some biology could be talked about that way, but not all of it. And trying to express biology in terms suited to chemistry would complicate biology to the point of incomprehension.

The same is the case with mathematics, for the reason you state. It's hard to see how physics would deal with abstract reasoning that didn't involve space and time, or why physics would make a better model for explaining mathematics than mathematics itself.


Thanks again for you view on these topics. They crop up frequently in philosophical discussions, and your post goes a long way toward clearing up the frequent confusion about them.

A very interesting subject. I'll try to lay some thoughts here.

I think mathematics is an indication of two things about us. First, the insane desire "to know" about reality. Second, the realization of the shortness of our senses to explore that reality. Math satisfies the previous two points perfectly because it's objective and based on the intellect.

That also makes me think it's universal. It might take different "representations", but its outcomes are indifferent.

I learned mathematics through Descarte!

According to a former teacher of mine, I do not have 'the mind for mathematics'. I can't say I disagree with his diagnosis....

Do you guys think that, ultimately, mathematics can be reduced to psychology?

If, after Kant, mathematics can no longer be ontologically grounded, as in some Platonic realm of forms or as the "language of nature," do its foundations become insecure and purely psychological?

And if they do, what then? Is it the case that we are destined to hit against the walls of our mind wherever we go, and mathematics is really just a rigorous formalization of the layout of those walls, just as logic is?

If the unique feature of humanity is externalized memory, then that would be logical.

Elena, a couple of thoughts on this. First, very few working mathematicians would be comfortable with the idea that they are doing nothing more than psychology. If anything, mathematicians tend to be realists of one sort or another. That's not dispositive, but its definitely something to take into consideration.

Second, in math disciplines, the quest for foundations tends to be circular. A book on number theory might decide to ground itself in set theory. Set theory in turn might seek some grounding in logic. But look at an introductory first order logic book, and you are likely to see the ideas grounded in arithmetic.

But, even if the foundations of math are "insecure," what difference does that make? I doubt that people will stop doing arithmetic simply because Russell and Whitehead failed to reduce it to first order logic.

This is a really interesting thread, but to comment even briefly on its many points is hardly possible, so I'll jump in at Elena's post 47. Duffy's assertion that the foundations "tend to be circular" is, I would say, really not true. There are different ways of approaching the foundations, but each can be built up consistently. Of course there are problems along the way: Russell's paradox, Godel's incompleteness theorem, but these reveal the essential nature of maths, rather than overturning its foundations.

(I'm from England: we say "maths" here rather than the American "math".)

The Russell-Whitehead project was actually very successful. But it doesn't end the story. In maths nothing ever does.

I did maths at University. In fact, maths was a consuming interest for me from ages 15 to 22. Later in life I tried to think about what maths actually is, and have never reached a conclusion. For a long time I saw it as purely man made; at others (like now) I see it as a discovery of real truths, though not usually of truths that can be observed anywhere in Nature.

Some comments on previous ideas in this thread: often there's the idea that maths is about numbers, counting, size, measuring. But that does not define it. Desargues' Theorem, for example, so fundamental as to be almost an axiom of projective geometry, has nothing to do with these concepts.

http://en.wikipedia.org/wiki/Desargue...

Another is that maths is the language of Nature (quantum theory and the like). To me, you can create mathematical allegories to parts of Nature, rather as Bunyan creates in The Pilgrim's Progress an allegory of the journey through life. But a lot of mathematical ideas only exist in mathematics. For example, the rational numbers can be put in 1-1 correspondence with the integers, the real numbers cannot. That is, the rationals are countable, the reals aren't. This is fundamental in modern maths. But it has no relevance to the Natural world, in which everything (presumably) is countable.

Elena, your question, "can maths be reduced to psychology?" The quick answer is "no" (I'm sure that's what Russell would have said), but could you elaborate a bit? I'm not quite sure what you had in mind there. I've never read Kant, but if he argues to divorce maths from Nature, I'd see him as correct. But "psychology" suggests that the key to understanding what maths is is to study human mental behaviour, and I can't see that getting far ...

P.S. I'm a different Martin from the other Martin

Two lovely (and well-known) quotes from Russell himself, in answer to the question of this thread,

"Pure Mathematics is the class of all propositions of the form "p implies q," where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants."

From The Principles of Mathematics (opening), and more whimsically,

"Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

Mysticism and Logic (Ch. 5)

Yes
I'm surprised you made that psychology comment, E

The Pthagorean theorem arises because every possible right angle with the sides measuring 3 and 4 units will have a hypotenuse of 5

This is inscripted in the ratios of geometry,
Independant of ANY state of consciousness

As you know,
I thing Math is indemic in Nature,
And not a human "invention"..

Rather i'd call it a "discovery"

Others may disagree

Martin, I realize that someone who is not familiar with the Kantian turn in philosophy may find the statement that mathematics could be treated as reducible to "psychology" strange. However, what a Kantian would mean by this is that we can no longer legitimately assume that any production of human reason (including mathematics) is a "reflection" of the language of nature, the universal mind, God, whatever. It is instead something that is constructed by the structures of rationality, and is therefore relative to those structures. Mathematics becomes epistemologically, not ontologically grounded. From this there is a short step to saying that it is psychologically, cognitively grounded only - I am basically asking if we have to make that step if we take the Kantian arguments seriously (and I cannot see how one could sidestep them, despite the flaws in his treatment of details). That being said, I don't think it's accurate to say that he divorced math from nature; instead he relativized and restricted its scope. I think his work had a more humble appraisal of the capacities of human reason than most realist philosophers are willing to entertain, who prefer to maintain a sort of closet Platonism on this issue and believe their work is a transcription of the "laws" or "language" of the universe. If one takes his philosophical arguments seriously, this opens a massive can of worms for human thought in general, as mathematics was traditionally considered our surest searchlight into reality.

Duffy, I realize that professional pride, as you point out, can be an obstacle to attaining philosophical clarity. People prefer to cloak themselves in various initiastic mystifications in order to deflect the validity of rational critiques made against the philosophical underpinnings of their work.To propose that mathematics might be only the most rigorous language of the human psyche, or the best formalization of the rules of thought, seems rationally plausible, albeit anathema as it threatens pet certainties.

I myself don't know. Something in nature -seems- to respond to this language of ours, but is it merely because we simply cannot go beyond its boundaries, and because we cannot but "see" the world through it? We will see confirmation of it everywhere because that is all we can do, but what can we -rationally- conclude from this limited confirmation?

Gun wrote: "Yes
I'm surprised you made that psychology comment, E

The Pthagorean theorem arises because every possible right angle with the sides measuring 3 and 4 units will have a hypotenuse of 5

This is i..."

Start at a point on the equator. Go 16000 miles due east, then turn due north and to 12000 miles. Your turn was a right angle. How far are you from your starting point. By the pythagorean theorem, the answer would be 20000 miles. But, if you travelled the shortest route over land, you would be much closer to 12,000 miles than 20,000 miles.

Another example. Go to a city that has a grid system: the EW blocks are streets, and the NS blocks are avenues. Ride down an avenue for 4 blocks, and then go up a street for 3 blocks. How far are you from your starting point. Not five blocks.

The "real" geometry that you insist upon is not always the most useful one. Moreover, it is one for which we only find very rough approximations in the real world. If you insist that there is some sort of Cartesian co-ordinate system that exists in the real world, instead of it being something that we impose on the real world with our imaginations, then I'm tempted to ask where the origin of that system is.

Just as I would not readily dismiss the mathematicians' intuition that math is somehow "real," I would also not lightly discard the notion that it is also a product of our minds, and not necessarily something we "find" in the world.

Martin wrote: "This is a really interesting thread, but to comment even briefly on its many points is hardly possible, so I'll jump in at Elena's post 47. Duffy's assertion that the foundations "tend to be circul..."

I should have been clearer when I said that "in math disciplines, the quest for foundations tends to be circular." What I meant is that I've observed math texts that attempt in their first chapters to ground the subject at hand upon some other field in math. And that tends to be circular. I see this as the various teachers in these fields tipping their cap to the general notion of needing a foundation, but there being no agreement on what field serves as the foundation, so they tend to run in circles. Since none of them need the foundations they supply, nor care about them very much, the exercise strikes me as being a little comical.

The goal of Principia Mathematica was to demonstrate that all math could be derived from first order logic. It was successful in that, with some uncomfortable back bending (like the theory of types), they derived proofs for much of simple arithmetic. But I don't see the incompleteness theorem as simply being a "problem along the way." It shows that the stated goal of the project was inevitably false, and thus, a failure. Your quote from Russell about the nature of math is quite elegant, and concisely, I think, states the conviction behind his and Whitehead's project. It also has been demonstrated to be false.

Well, I give up trying to understand this Kantian stuff.

But to return to Duffy's point, I understand that one math text might define A in terms of B, and another might define B in terms of A, and this would seem to be circular. I was merely stressing that it does not have to be ... so there is no real disagreement here I think. Sorry.

(Of course deriving A from B, and B from A can be really interesting as a way of showing the equivalence of two axiom systems.)

Regarding Princ. Math., I guess it depends on what the imagined goal was. I have always taken it to be just the derivation of maths from a small number of logical primitives. (All credit to Russell to realise that naive set theory was not adequate for that.)