[drewfactor]

I'm currently reading a book about the philosophy of mathematics. It's an interesting read and although I don't fully understand all of the notation and proofs, the general idea is that Platonic realism is the only fully defensible position to take on the philosophy of math. The general premise is that Mathematics gives us a view into the Platonic realm, that exists independant of us, and provides a representation of reality that is comprised of sense data. The author (I think) correctly discredits the nominalist and constructivist views of mathematics.

My question is what would an Objectivist view on the philosophy of math be? Based on my understading of Objectivist epistemology, math is not some form of "insight" via the "minds eye" as the author of my book keeps saying, but math begins as an inductive process and through measurement omission mathematical concepts arise just as all other concepts arise.

Is anyone familiar with an Objectivist conception of the philosophy of mathematics?

[DPW]

I'm not a mathematician, nor have I studied that matter in depth, but for what it's worth: mathematics is the science of measurement. It exists to work out the process by which we perform actual measurements. The way to keep it reality oriented, therefore, is to know what kind of things in reality it's used to measure. In other words, you must keep in mind its applications.

[Nate T.]

drewfactor, on Oct 21 2005, 07:20 PM, said:

Is anyone familiar with an Objectivist conception of the philosophy of mathematics?

As an aspiring math professor, these kinds of questions have interested me before, so I've given them some thought. The short answer is: you're right. Mathematicians of the Platonist persuasion are intrinsicists-- they believe that the concepts with which they work have existence independent of human minds.

As in many other fields dealing with concepts, the philosophy of mathematics has generally split along the intrinsic(Platonist)/subjective(Formalist) false dichotomy. Constructivism, OTOH, is an attempt to answer troubling paradoxes resulting in the use of the Axiom of Choice and applying the Law of the Excluded Middle to infinite sets, adn so addresses different questions than the other two I mentioned. IMO, constructivism is far too strict and too vague in what it considers "constructible."

So, while there is no Objectiviost philosophy of mathematics, I don't think there's really a pressing need for one, since the biggest problem in the Philosophy of mathematics has been the "ontological" status of mathematics concepts, which Rand already dealt with in IOE.

[drewfactor]

DPW: I agree. I think your point is true that applications are necessary in order to keep mathematics reality oriented. The thing is, though, the mathematician author of my book and my brother who is studying senior level mathematics and engineering seem to be driving at a point. They seem to emphasize the need to keep mathematics detached from reality in order to advance it as a science/art/discipline. My brother started in Engineering and has all but abandoned that field to pursue studies in the pure math. I don't understand much of it, but it almost has this mystical quality about it; almost meditative for him. I get the impression that holding the platonic view of math (ie. emancipating your thoughts from the constraints of reality) enables discoveries in the field that may have applications *later*. An example of this appears to be in the area of set theory.

Nate: It seems like the false dichotomy between the intrincisist and subjectivist camp are certainly due to the ontological nature of mathematical concepts. Basically, the intrincisist is dogmatically asserting that the concept has existence independent of us (regardless of the role consciousness plays in the formation of the concept) and the subjectivist is replying that, no, the concept is basically an arbitrary human creation. I think I'm starting to see how the Objectivist position fits in. They are both right and both wrong in many respects.

[DPW]

drewfactor, on Oct 21 2005, 08:33 PM, said:

DPW: I agree. I think your point is true that applications are necessary in order to keep mathematics reality oriented. The thing is, though, the mathematician author of my book and my brother who is studying senior level mathematics and engineering seem to be driving at a point. They seem to emphasize the need to keep mathematics detached from reality in order to advance it as a science/art/discipline. My brother started in Engineering and has all but abandoned that field to pursue studies in the pure math. I don't understand much of it, but it almost has this mystical quality about it; almost meditative for him. I get the impression that holding the platonic view of math (ie. emancipating your thoughts from the constraints of reality) enables discoveries in the field that may have applications *later*. An example of this appears to be in the area of set theory.

Oh, well that's just the division of labor and that's completely proper. There is nothing wrong if your own focus is on theoretical mathematics, just as there is nothing wrong if your focus is on theoretical science. The point is that a theoretician must never forget that the value of his work is in its application to human purposes -- to man's life. Most importantly, he must not oppose the application of his work to human life.

But this does raise an interesting question: how can a theoretical mathematician, who is working on math that does not yet have an application in reality, stay reality oriented? There I don't have a good answer. This doesn't come up in other theoretical sciences -- theoretical physics is not theoretical in the same sense that math is. It is about reality, not about our means of knowing (i.e., measuring) reality.

The best advice I can offer is just of the general sort I've already indicated. He must keep in mind that math is not a platonic game but the science of measurement. He must keep in mind the ultimate purpose of his work (even if he doesn't carry out this purpose) is the application of his work to reality. If he does that, I don't think he's doomed to rationalism.

This post has been edited by DPW: 21 October 2005 - 07:51 PM

[dondigitalia]

DPW, on Oct 21 2005, 08:49 PM, said:

But this does raise an interesting question: how can a theoretical mathematician, who is working on math that does not yet have an application in reality, stay reality oriented?

I can't believe I didn't notice this thread before, since this is one of my main areas of interest in Philosophy.

Don's question is really the most fundamental point, so that's where I'll start. Fundamentally, mathematics is concerned only with metaphysics and epistemology, and between the two, mostly with epistemology (although, of course, the metaphysics HAS to be there first). So the question that needs to be asked here is: How do we keep epistemology reality-oriented? The answer is in the Objectivist theory of concept-formation.

Mathematics, in some respects, is a reversal of what we normally do in epistemology. For instance, in a normal concept, we retain characteristics, but omit the particular measurements. However, in forming numerical concepts, which lie at the root of mathematics, we retain measurements, but omit the particular characteristics. There is a whole hierarchy of how mathematical concepts are derived from this one principal, which I won't go into in detail, but basically, there we can't arrive at the concept of 1 until we have concepts for a number of other numbers, and we can't arrive at the concept 0 until we have 1 and a number of natural numbers, negative numbers come after 0, rational numbers come after natural numbers, etc. There's a whole chain of abstraction, including the abstraction of mathematical methods from actions performed using those numbers.

Another way in which mathematics is a reversal is that, in normal thinking, we arrive at our premises by a process of induction, and then apply those premises by deduction. In mathematics, we deduce our premises from a set of axioms, and then apply them by induction.

In answer to Drew's first question: Nobody is familiar with an Objectivist Philosophy of Mathematics, because no such thing exists. What little Ayn Rand said on the subject of mathematics was within the context of epistemology proper, and wasn't really a Philosophy of Mathematics. That said, the above ideas are in no way representative of Objectivism; they are a general overview of my own thoughts on the subject. I have no plans to go into a full in-depth explanation at the present time. I may be willing to answer some very general questions on the Philosophy of Mathematics, however.

[peoplater]

Everyone, everyone, can I have your attention please. All philosophers of mathematics and science in general can stop asking any more questions because I have figured it out. It is impossible to do math or physics or chemistry or anything else without having it in some way grounded in reality. So it does not matter if you are an objectivisit or a platonist or anything else, if you are doing math then your intuition is necessarily grounded in reality.

[Lord Poppycock]

[BurgessLau]

peoplater, on Nov 12 2005, 01:00 AM, said:

So it does not matter if you are an objectivisit or a platonist or anything else, if you are doing math then your intuition is necessarily grounded in reality.

Two questions arise for me. First, what do you mean by "objectivist" and how does that differ from "Objectivist"? Second, what do you mean by "intuition" and how does your meaning differ from the way the term is used in Objectivism? (See "Mysticism," The Ayn Rand Lexicon, first paragraph.)

[Thales]

drewfactor, on Oct 21 2005, 06:20 PM, said:

Is anyone familiar with an Objectivist conception of the philosophy of mathematics?

Have you seen Ron Pistaro's articles on this matter? He shows how to derive simple mathematical concepts by an inductive approach. He goes through numbers and various mathematical operations. These are at a basic level, but that's probably the best place to start!


Check out The Intellectual Activist back issues. You'll find the following:

Vol. 8, No. 4, Jul 1994. The Foundation of Mathematics, Part 1, by Ronald Pisaturo and Glenn D. Marcus.

Vol. 8, No. 5, Sep 1994. The Foundation of Mathematics, Part 2, by Ronald Pisaturo and Glenn D. Marcus.

Vol. 12, No. 10, Oct 1998. Mathematics in One Lesson, Conclusion, by Ronald Pisaturo.

Vol. 12., No. 9, Sep 1998. Mathematics in One Lesson, Part 1, by Ronald Pisaturo.



The following aren't on the derivation of mathematical concepts, but are additional articles by him on how math has been under assault:

Vol. 14, No. 10, Oct 2000. Undermining Reason: The 20th Century's Assault on the Philosophy of Mathematics, Part 1, by Ronald Pisaturo.

Vol. 15, No. 11, Nov 2001. Undermining Reason: The Assult on the Philosophy of Mathematics, Part 2, by Ronald Pisaturo.

Vol. 15, No. 12, Dec 2001. Undermining Reason: The Assult of the Philosophy of Mathematics, Conclusion, by Ronald Pisaturo.


He also used to have a taped lecture available from the Ayn Rand Bookstore. I'm guessing it's still there!

[peoplater]

BurgessLau, on Nov 12 2005, 08:58 AM, said:

Two questions arise for me. First, what do you mean by "objectivist" and how does that differ from "Objectivist"? Second, what do you mean by "intuition" and how does your meaning differ from the way the term is used in Objectivism? (See "Mysticism," The Ayn Rand Lexicon, first paragraph.)

It doesn't differ in any way from Objectivist, at least I don't think it does. By intuition I mean the images that are conjured up when you see a+b=c. Even the greatest of mathematicians do not think in totaly abstract terms. There is some image in their minds when they are trying to prove or disprove something, and that image does not just come out of nowhere. It comes from their years of experience in life and in mathematics. One more point. The problem is that it is philosophers that do philosophy of mathematics and not mathematicians. To speak about mathematics one should at least know enough to understand some non-trivial results in algebra and topology, but I do not think there are any such philosophers, and most mathematicians do not need philosophical justification to do what they do.

This post has been edited by peoplater: 12 November 2005 - 02:48 PM

[BurgessLau]

Peoplater,

1. OBJECTIVISM. I have seen the term "objectivism" used occasionally in the history of philosophy to refer to a particular idea: "Things exist independently of the mind." I recall, but not with certainty, that I saw that usage listed in the 10-volume Routledge Encyclopedia of Philosophy, in the "objectivism" article. The main point here is that it refers to one idea.

On the other hand, "Objectivism" is a proper name. It is the label for a whole philosophy, the one that Ayn Rand created. For cognitive clarity, as well as out of respect to its creator, "Objectivism" deserves an uppercase "O." Also, the last time I checked, the Forum Rules prohibit egregious misspellings and cite "objectivism" (and, by implication, its variants) as an example.

2. INTUITION. In Objectivism, which is the philosophy that sets the context for ObjectivismOnline.net, "intuition" refers to an act of mysticism -- which Objectivism rejects totally, in favor of reason.

I was hoping for a definition of intuition from you. Instead you offered an example of an alleged intuition: "the images that are conjured up when you see a+b=c." I am unsure what you are saying. Perhaps you will elaborate. For example, what images does "a" conjure up?

In the meantime, I would suggest that thinking in images is impossible. I -- and so far as I know, everyone else -- can think only in symbols, which are usually words but could be numbers or other symbols, I suppose. (I am not a mathematician.) Ayn Rand discusses the epistemological role of words in Introduction to Objectivist Epistemology, pp. 10-11, 40, and 163-177.

For more on the error of attempting to think in images (which is a contradiction in terms), see the recently published Ayn Rand Answers: The Best of Her Q&A, edited by Robert Mayhew, available from The Ayn Rand Bookstore. On pp. 177-178, Ayn Rand informally answers this question: "Is it possible to think in images, rather than with words?"

Also, I recall that Leonard Peikoff spoke to the Ford Hall Forum five or so years ago about this very subject, the error of trying to think in images. His lecture is available through ARB as a recording. (I have not heard the lecture, but perhaps I read a print version of it in The Intellectual Activist.)

Working mentally with images -- creating them, manipulating them -- is imagination. By contrast, thinking is working with symbols, particularly words, as labels for concepts that refer to things in reality; or, in the case of proper names, the words refer directly to things themselves (as in "Mars"). Using those symbols, a thought is a statement about some aspect of reality: "Man is a rational animal." An image is a picture of some aspect of reality -- for example, a mental picture of a particular gorgeous man or woman.

This post has been edited by BurgessLau: 13 November 2005 - 07:52 AM

[Hal]

BurgessLau, on Nov 13 2005, 07:36 AM, said:

In the meantime, I would suggest that thinking in images is impossible.

I disagree; I often use 'imagistic intuition' when solving math problems, and many mathematicians report the same. Sometimes the feeling of what the answer should look like occurs well before youve worked out how to actually reach it. There's a flash where you somehow 'see' the answer, even though you cant quite formulate the route which brings you there. This kind of thing is very hard to describe though, since youre essentially trying to descibe a thought process using words. I'd perhaps compare it to trying to remember what a particular word for something is - you have some vague idea what the 'shape' of the word should be, even though you cant quite remember the word itself. Its not really like that of course, but its a reasonable analogy.

But, assuming that your claim isnt purely linguistic (ie a statement about the use of the word 'thinking'), it would require some kind of psychological evidence rather than purely philosophical argument.

On a sidenote

Quote

Tammet is calculating 377 multiplied by 795. Actually, he isn't "calculating": there is nothing conscious about what he is doing. He arrives at the answer instantly. Since his epileptic fit, he has been able to see numbers as shapes, colours and textures. The number two, for instance, is a motion, and five is a clap of thunder. "When I multiply numbers together, I see two shapes. The image starts to change and evolve, and a third shape emerges. That's the answer. It's mental imagery. It's like maths without having to think."

...

Last year Tammet broke the European record for recalling pi, the mathematical constant, to the furthest decimal point. He found it easy, he says, because he didn't even have to "think". To him, pi isn't an abstract set of digits; it's a visual story, a film projected in front of his eyes. He learnt the number forwards and backwards and, last year, spent five hours recalling it in front of an adjudicator. He wanted to prove a point. "I memorised pi to 22,514 decimal places, and I am technically disabled. I just wanted to show people that disability needn't get in the way."

http://www.guardian....1409903,00.html



edit: A more down to earth example would be something like the experiments carried out by Roger Shepard, which involved showing people a pair of 3-d shapes that were rotated in relation to each other, and asking them whether the 2 images depicted the same shape, or whether they were in fact mirror images. Most people responded that they solved the problem by conjuring up mental pictures of the shapes, and rotating them in their "mind's eye".

A similar experiment is described here (although i dont know the full details of this one)

Quote

Stephen Kosslyn (1980) and his colleagues have documented similar results concerning people's abilities to imagine maps. In a typical experiment, subjects are shown maps of a fictional island with some marked locations: a tree, a house, a bay, etc. The maps are removed and the subjects are then asked to focus mentally on one location on the map and then move their attention to a second location. The finding was that the time it takes to mentally scan from one location to the other is again a linear function of the distance between the two positions on the map. The interpretation is that the subjects are scanning a mental map, in the same manner as they would scan a physically presented map.

edit 2: how do you suppose that people who only know a sign language think? Or someone like Helen Keller who was born both deaf and blind, yet reported that she had 'thought processes' before she managed to learn a language? These are highly interesting questions which have nothing to do with philosophy.

This post has been edited by Hal: 13 November 2005 - 11:40 AM

[Hal]

Thales, on Nov 12 2005, 01:03 PM, said:

Have you seen Ron Pistaro's articles on this matter?

Are any of these available online, or have they published in an academic journal which I can access using athens (eg one archived on JSTOR)? I'm curious what sort of arguments he's using, but dont really want to pay money for something when I have no idea what the quality will be like...

peoplater, on Nov 12 2005, 03:32 PM, said:

The problem is that it is philosophers that do philosophy of mathematics and not mathematicians.

This isnt true at all; most influential philosophers of mathematics have either been prominent mathematicians, or highly acquainted with the subject material. Names like Hilbert, Brouwer, Kronecker, Cantor, Poincare, Polya and Frege spring to mind.

This post has been edited by Hal: 13 November 2005 - 11:50 AM

[Hal]

dondigitalia, on Oct 26 2005, 01:22 AM, said:

Another way in which mathematics is a reversal is that, in normal thinking, we arrive at our premises by a process of induction, and then apply those premises by deduction. In mathematics, we deduce our premises from a set of axioms, and then apply them by induction.

Well, you have to ask where the axioms originally come from. Normally a branch of mathematics is only axiomised once it is fairly mature - when you have a large number of results, people will start to see if there is some minimum number of basic statements they can all be deduced from, and these will become the standard axioms. The way that mathematics is presentd in textbooks isnt the way which it actually evolves - mathematicians do not generally start with a set of axioms and try to formally deduce things from them. Euclid's elements would be a classic example of this; most of the results contained in it were already known, although it was Euclid who first had the idea of reducing them to their basic premises. A more modern example would be something like group theory - the formalised notion of (eg) groups and ideals were postulated after many key results had already been found.

You can also have disagreements about which axiom sets should be used - different axioms will have different consequences, so people can disagree about the best abstraction in a pariticular case. For instance, there was (afaik) disagreement about what axioms best captured our intuitive notion of a metric space, with the result that 2 competing axioms sets were used. One of these is now the commonly accepted axioms for metric spaces, while the other has been adapted for pseudometric spaces, which are slightly different. The Peano axioms for the natural numbers would be another example - there are certain results of number theory which the paeno axioms are not strong enough to prove, which slightly diminished their utility.


So yeah, mathematics isnt _that_ different from what we normally do. You postulate a certain axiom system which you intend to capture certain results, then you check what consequences it has, and if you dont like them, you adjust your axioms till they work.

Quote

There is a whole hierarchy of how mathematical concepts are derived from this one principal, which I won't go into in detail, but basically, there we can't arrive at the concept of 1 until we have concepts for a number of other numbers, and we can't arrive at the concept 0 until we have 1 and a number of natural numbers, negative numbers come after 0, rational numbers come after natural numbers, etc.

This is true psychologically, but not logically. Yes, its a psychological fact about humans that we learn the concept of the natural numbers before that of 0. But this doesnt imply that the natural numbers come first from a mathematical point of view; the positive integers are normally defined in terms of the empty set with zero coming first: 0 = {}, 1 = {0} = {{}}, 2 = {0,1} = {{}, {{}}} and so on.

Rational numbers do come after natural numbers in the standard constuction, since they are defined as equivalence classes of ordered pairs of integers, but there are different constructions where this is not true. For instance, when you construct your number system using something called surreal numbers, you start by creating a particular class of rational numbers (dyadic fractions) which you can then use to define the integers (and later, the real numbers).

This post has been edited by Hal: 13 November 2005 - 12:40 PM

[Thales]

Hal, on Nov 13 2005, 12:47 PM, said:

Are any of these available online, or have they published in an academic journal which I can access using athens (eg one archived on JSTOR)? I'm curious what sort of arguments he's using, but dont really want to pay money for something when I have no idea what the quality will be like...

They are only available from The Intellectual Activist so far as I know. $4.00 per issue according to the website link. The articles are directed toward a general Objectivist audience, but would be beneficial to anyone interested in mathematics.

What Pisaturo does is apply the Objectivist epistemology toward deriving mathematical concepts. He starts with counting, the most basic mathematical concept, and shows how this is developed inductively. He covers concepts like multiplication, exponents, irrational numbers. He covers algebra and geometry, etc. He doesn't cover them in depth, mind you, these are just articles, but they give you grounding in his approach. Pisaturo has a degree in mathematics from MIT. Marcus is/was a professor of mathematics. So, he's qualified on that front and he's also a knowledgeable Objectivist.

This is not your standard mathematical approach. It's not rationalistic. So, this is where the value lies in reading it.

[Hal]

Thales, on Nov 13 2005, 02:19 PM, said:

Pisaturo has a degree in mathematics from MIT. Marcus is/was a professor of mathematics.

Well thats certainly encouraging; I might order one of the backissues. The problem I've always found in the past with attempts to 'construct mathematics rationally' (or whatever) is that they tend to be exercises in psychologism, largely focusing on how humans acquire knowledge of mathematical concepts. And while that can be interesting from the point of view of cognitive/developmental psychology, it has very little to do with either mathematics or logic.

This post has been edited by Hal: 13 November 2005 - 02:14 PM

[Thales]

Hal, on Nov 13 2005, 03:11 PM, said:

Well thats certainly encouraging; I might order one of the backissues. The problem I've always found in the past with attempts to 'construct mathematics rationally' (or whatever) is that they tend to be exercises in psychologism, largely focusing on how humans acquire knowledge of mathematical concepts. And while that can be interesting from the point of view of cognitive/developmental psychology, it has very little to do with either mathematics or logic.

Keep in mind that this is induction. All knowledge is ultimately derived inductively, including mathematical knowledge. This is why it has everything to do with math and logic.

I think it will give you some insights.

[peoplater]

I very much agree with most of what Hal has said and on the issue of intuition and imagination in solving mathematical problems there is a great book by J. Hadamard titled "The Psychology of Invention in the Mathematical Field". The author discusses what happens in his mind when he is doing mathematics and compares his thought process to the thought processes of other mathematicians. Not surprisingly many mathematicians think in vague images, very few of them think in exact symbolic terms. Most of them even admit to just finding the solution of a problem just out of nowhere, after working in it for a very long time. That is why it seems to me to be pointless to subscribe to any kind of philosophy of mathematics and ask questions like "what is a number?", "where does it come from?" and so on. All mathematical things are just a byproduct of the way our minds are put together. The idea of a number is inherent in the construction of the brain. I do not think it can be described in any other terms.

[LauricAcid]

peoplater, on Nov 12 2005, 07:02 PM, said:

it is philosophers that do philosophy of mathematics and not mathematicians.

That is a remarkably false statement.

peoplater, on Nov 12 2005, 07:02 PM, said:

understand some non-trivial results in algebra and topology, but I do not think there are any such philosophers

What philosopher of mathematics do you point to as one not knowing basic abstract algebra and topology?

peoplater, on Nov 12 2005, 07:02 PM, said:

To speak about mathematics one should at least know [...]

To make vast generalizations about philosophers of mathematics, one should at least know of one.