Talk:Mathematics/Archive 8

Megalithic mathematics

 * ''Neolithic monuments on the British Isles are constructed using Pythagorean triples.

This, at a minimum, requires a source. There is a long-standing conjecture that Neolithic peoples could have used Pythagorean triples for a right angle; it was stated about Egypt, and I believe Morris Klein is responsible for it. I know of no evidence for it anywhere, and the British Isles is a most unlikely place to find some: ropes will not survive, and few of the British megaliths have right angles. Septentrionalis 17:14, 1 June 2006 (UTC)

Applied mathematics
The list under this heading seems a bit biased toward areas of application (e.g., Mathematical physics, Mathematical economics) rather than methods of applied math. (e.g., Representation theory, differential equations, approximation theory). To an extent this is also true of things listed at applied mathematics. Some fields, like fluid dynamics, are in both categories. But I think it fits in more with the other headings to list subfields of math. like ODEs, PDEs, etc. preferentially over Mathematical biology and such, or to list more of the subfields at least. JJL 21:22, 1 June 2006 (UTC)


 * Maybe part of the reason for this is that it is slightly ridiculous to refer to representation theory as applied maths, even though it is used in mathematical physics. JPD (talk) 09:55, 2 June 2006 (UTC)

Could it be that JJL was talking about some Branches of representation theory ? Brian W 00:19, 3 June 2006 (UTC)


 * Well, I don't see JLL referring to representation theory as applied mathematics, just as a method of it. That's certainly a reasonable statement.  Even if one excludes what you might consider rather "esoteric" applications in mathematical physics, there are still plenty of applications to very concrete situations involving cryptography, crystallography, error-correcting codes, etc.  --C S (Talk) 10:54, 3 June 2006 (UTC)


 * I see putting representation theory under the heading Applied mathematics as referring to it as applied mathematics. My point was that while something like mathematical physics may be appropriate under such a heading, it wouldn't be right to put representation theory as a whole there. (Representation theory as a whole is not represented by that link at the moment, either!) Anyway, I think the reason the list is "biased" to areas of application, is because the methods are also methods/areas of pure maths. I did not mean to imply that mathematical physics is the only application of representation theory, and the question of whether cryptography, error-correcting codes or even mathematical physics are considered "applied maths" is another issue altogether. It would probably be a good idea to put differential equations in there, however. JPD (talk) 15:16, 3 June 2006 (UTC)


 * I don't care about rep. theory (which I assumed, without checking, would go to a subset of approx. theory--I was thinking of Fourier series representations, not group theory) in particular. Clearly, as it stands it should not be in applied math. I meant more about methods/areas like approx. theory vice areas of applicability like math. bio.--applied math. over applicable math. JJL 05:01, 5 June 2006 (UTC)


 * Sounds like a good idea to me. JPD (talk) 10:23, 5 June 2006 (UTC)

Jargon
The section Notation, language, and rigor talks about mathematical jargon, but as has been mentioned, this refers to words. Perhaps code would be more appropriate to refer to the notation itself, or perhaps language. I feel this sentence stops a bit abruptly with music, and would like to bring it back to finish with something about the mathematical notation. Stephen B Streater 11:43, 4 June 2006 (UTC)

New intro
Looks good to me. Rick Norwood 16:59, 4 June 2006 (UTC)
 * Yes I like the new intro as well. I've been away since Thursday in New York, glad to see we've made some progress here ;-) Paul August &#9742; 20:38, 4 June 2006 (UTC)


 * I too have been away this weekend in NY, and I too like the new intro. As to discussions further down the page such as concepts vs. topics, I don't much care, but "discipline encompassing the study of concepts" does seem a bit wordy to me. As to the fact that it doesn't clearly state that the results of such studies may be put to practical use, that doesn't bother even an applied mathematician like me! JJL 14:49, 6 June 2006 (UTC)

Congrats, this time the intro looks very good (oh yes, I am watching you all since a long time). Now, it's time to do some work over the logic related articles. Brian W 20:53, 4 June 2006 (UTC)
 * Wow, the new intro is worlds better. &mdash;M e ts501 talk 20:59, 4 June 2006 (UTC)

Well done everyone! -- darkliight [&pi;alk] 01:15, 5 June 2006 (UTC)

RE: Opening paragraph. Good grammar can be as precise and pleasing as good notation. We should try to be consistent in use of either nouns, or present participles in series. Being good in both enhances clarity. Bcameron54 01:40, 5 June 2006 (UTC)


 * Hi there. I like the opening paragraph as well, with one reservation.  Could we change the word "topics" to "concepts"?  I generally think of a "topic" as something which might be covered by a textbook, chapter, or a class (such as geometry or group theory).  Quantity, structure, space and change are general concepts which are used in a variety of topics.  I didn't want to make the change without getting some feedback first though, so if I hear nothing back for a day or so I'll go ahead and do it.  capitalist 02:24, 5 June 2006 (UTC)


 * I agree, "concepts" is better. -- Jitse Niesen (talk) 04:02, 5 June 2006 (UTC)
 * Comrade Jitse, the comissar is watching! No, we don't agree with those capitalists/monoplists/bourjois! Not at all. Oleg Alexandrov (talk) 04:14, 5 June 2006 (UTC)
 * Bah! Then I will now launch a hostile takeover!  I will buy Wikipedia!  Oh, wait.  My board of directors has just voted me out.  Never mind.  :0)  capitalist 04:11, 6 June 2006 (UTC)
 * Agreed here too. I made the change, but needed to add the words the study of to make the sentence flow. If anyone disagrees with that feel free to remove them. -- darkliight [&pi;alk] 04:51, 5 June 2006 (UTC)
 * I think restricting it to study is a bit odd because it excludes use. A musician plays music as well as writing it. But no one else seems to be concerned, so I'll leave it until I have a better argument. Stephen B Streater 06:52, 6 June 2006 (UTC)
 * Music is the study of notes. But what about actually playing the music? Mathematics is the study of quantity. See my point? Stephen B Streater 08:30, 6 June 2006 (UTC)
 * Mmm, maybe I'm not seeing your point, but to be fair, we're not describing a mathematician or the use of mathematics, we're attempting to define the discipline of mathematics. Our opening sentence is, to the best of our collective ability, a definition of mathematics. We've said what we think it is. Now, a mathematician certainly uses mathematics, and this should be mentioned throughout the article, but the use of mathematics in the definition of mathematics is not required, surely? -- darkliight [&pi;alk] 10:49, 6 June 2006 (UTC)
 * The word "study" has connotations of learning. To use your point, mathematicians study topics just as much as they use them, but you don't seen to mind the mention of studying. I would prefer a less restrictive form in the definition, such as "analysis of". This includes learning and use. Stephen B Streater
 * For me the connotation is not that strong. To solve a problem, unless it is routine, I study it. I study the issues and I study the approaches, for doing which I may repair to my study room. (Maybe I'm taking "mathema" too literally. :) ) --Lambiam Talk 12:37, 6 June 2006 (UTC)
 * Suppose you are solving the equation 2x=4. Would you say this involved study? Study to me would be learning or working out how to solve the equation ax4+bx3+cx2dx+e=0. But applying the formula would not be study, but would still be mathematics. Stephen B Streater 13:57, 6 June 2006 (UTC)
 * What's the difference between the two solutions, other than length? -lethe talk [ +] 14:52, 6 June 2006 (UTC)
 * I don't see applying a formula as study, but I do see it as mathematics. A random dictionary here (v1.0.1 of Dictionary on my Mac) says: study is "the devotion of time and attention to acquiring knowledge on an academic subject". On this definition, study excludes using the knowledge. I think mathematics encompasses the use of mathematical knowledge, not just learning it. So applying a formula is Mathematics as much as studying it is. Stephen B Streater 15:25, 6 June 2006 (UTC)
 * The above was definition 1. Definition 2 gives: "a detailed investigation and analysis of a subject or situation" - again not actually applying knowledge. Stephen B Streater 15:31, 6 June 2006 (UTC)
 * While I'm here, it gives mathematics as: "the abstract science of number, quantity, and space. Mathematics may be studied in its own right ( pure mathematics), or as it is applied to other disciplines such as physics and engineering ( applied mathematics). • [often treated as pl. ] the mathematical aspects of something : the mathematics of general relativity." - again not restricted to the study of things. Stephen B Streater 15:33, 6 June 2006 (UTC)
 * I'll just mention that I'm prepared to change my view if that's not how everyone else sees it. Stephen B Streater 15:36, 6 June 2006 (UTC)

The word "mathematics", especially in the abbreviated form "math" or "maths", is both the study of mathematics and also a subject taught in school. The same is true of "history", but we expect an encyclopedia article on history to focus on the professional meaning, not the schoolboy meaning. Solving 2x=4 is something a high school student would call "math" but an encyclopedia article will naturally focus on the more professional meaning. Rick Norwood 15:07, 6 June 2006 (UTC)

I'm weighing in on the side of removing the phrase "the study of". Does a discipline encompass concepts or does it encompass the study of those concepts (or both)? I think a discipline encompasses concepts. Mathematics is about quantity; it is not about the study of quantity. A discipline about the study of quantity would involve concepts such as how people learn, how they form abstractions or whatever. So a discipline like Cognitive Theory might encompass the study of quantity or the study of cells. But Mathematics encompasses the concept of quantity and Biology encompasses the concept of cells. Then again, I could be just babbling again... capitalist 02:37, 7 June 2006 (UTC)


 * We could say : "is a discipline that deals with concepts such as ...". The articles Physics, Chemistry and Biology all have "deals with". --Lambiam Talk 04:15, 7 June 2006 (UTC)
 * This meets my constraints. I would also be happy with removing "the study of" completely, as this is more concise and is more accurate to me, in that mathematics doesn't "do" anything, it's more just sits there encompassing various areas. It's mathematicians who deal with things. I think it was changed from this, though I didn't catch the reason. "Mathematics is a discipline encompassing concepts such as ..." Stephen B Streater 06:10, 7 June 2006 (UTC)


 * Matching the sciences articles sounds good to me. JJL 12:50, 7 June 2006 (UTC)

It looks good, everyone looks happy with it and so I was bold and changed it ... again :) -- darkliight [&pi;alk] 16:48, 9 June 2006 (UTC)


 * Thanks. Stephen B Streater 17:21, 9 June 2006 (UTC)

More intro work.
Since we're on a bit of a roll with the intro, here are a few more things I think are worth discussing: Anyway, I'm just trying to keep the ball rolling. Cheers -- darkliight [&pi;alk] 08:29, 17 June 2006 (UTC)
 * Should the word results be replaced with ideas or something similar in the last line of the first paragraph?
 * Since the last line of the first paragraph is alluding to proof, I thought this sentence could be expanded a bit suggesting that ideas can sometimes take hundreds of years to prove, with many old ideas still yet to be proven and many new ideas still coming in. I'm not sure how to word it exactly, but I think it would help people realise that mathematics is a still growing body with many unsolved problems remaining.
 * Ideas for the second paragraph? I was thinking this paragraph could be used to introduce the history of mathematics and lead into the third paragraph.
 * Third paragraph? I was thinking this could be used to describe the current state and current uses of mathematics, similar to what the second paragraph does now.
 * Any ideas on what to do with the line about the abbreviation of mathematics? I don't think it warrants its own paragraph, but I don't think it really belongs in the etymology section either .. or does it?


 * A quick reaction to the very last bit: both Math and Maths redirect to Mathematics, and it is customary to mention commonly used alternative designations in bold in the intro (usually even in the very first sentence, but in this case that is not a good idea). --Lambiam Talk 11:10, 17 June 2006 (UTC)


 * I agree with Lambiam, we need to keep the brief mention of "math" and "maths" where it is. Darklight's other ideas sound good, but of course the devil is in the details.  Why not try making the changes one at a time and see what happens? Rick Norwood 13:02, 17 June 2006 (UTC)


 * I'm not too keen on ideas. You cannot prove an idea, you have to formulate it first. However, I agree that results is not good either when talking about statements that are not yet proven.
 * I am curious how the history can be summarized in one paragraph, and I'm not sure that would be sufficiently important to put in at the top. But it may well turn out to be better than I imagine. Perhaps you can write a rough start to give us an idea?
 * I'd love to see the "math" / "maths" paragraph go. It is indeed customary to mention other terms. However, that is so that readers will not arrive at the page in bewilderment. For instance, if you type in Burma, you end up at Myanmar, and if you don't know that these names refer to the same country and it isn't mentioned in the article, you'll be very confused. I don't think the possibility for confusion is great in the math(s) / mathematics case. -- Jitse Niesen (talk) 13:55, 17 June 2006 (UTC)

A friend and I had an attempt an the last sentence:

Mathematics is the discipline that deals with notions such as quantity, structure, space and change. It evolved, through the use of abstraction and logical reasoning, from counting, calculation, measurement and the study of the shapes and motions of physical objects. Mathematicians aim to justify/verify their concepts by rigorous deduction from axioms and definitions, which in some cases can take hundreds of years to accomplish.

I tried keeping changes to a minimum, my main goal here removing the word results and hinting that mathematics is not complete. Ofcourse these are just some ideas for now, and I'll write something up for the second and third paragraphs next week. Cheers -- darkliight [&pi;alk] 10:28, 19 June 2006 (UTC)
 * Instead of ideas, we used the word concept in place of the word results and ideas. It seemed to fit well, but the word was already used in the first paragraph. Another version I thought of was, reading almost straight of Jitse's comment: Mathematicians aim to formulate and justify/verify their ideas by rigorous deduction from axioms and definitions, which in some cases can take hundreds of years to accomplish.
 * To save confusion we changed concepts in the first paragraph to notions.
 * What do people think about the word verify in place of justify? While they both work, I think justify is little neater ... it has a more of a creative aspect to it and doesn't have the connotations of plugging numbers into a formula to get your answer.
 * To convey that mathematics is still a progressing discipline, we've simply added which in some cases can take hundreds of years to accomplish. to the end of the last sentence. We could also add with many ideas still unproven or unjustified or unverified, but it starts to drag along a bit.


 * I suggested verify instead of justify because verify is a more precise, objective, specialised and idealised form of justify, and connotes truth (being derived from the Latin word for true). While the ideal may not be realised as frequently as we would like, an introductory paragraph should be as idealised as possible.
 * I also proposed somehow introducing the word explore, to suggest both the creativity and ongoing evolution of mathematics. My poor sleep-deprived brain came up with a somewhat clunky Mathematicians use rigorous deduction from axioms and definitions to explore and verify these concepts, a process which in some cases can take hundreds of years. Perhaps someone can work this into a prettier sentence.
 * -- Cuadan 13:04, 19 June 2006 (UTC)


 * I agree with Darkliight's dislike of verify as suggesting some trivial activity. Replacing idea with concept does not help with the problem I stated above: what does it mean to verify/justify concepts? However, there are some good ideas. How about:
 * &hellip; Mathematicians explore these and related concepts, aiming to formulate statements and establish their truth by rigorous deduction from axioms and definitions. This process may take hundreds of years.
 * To be honest, I'm not so happy with the last sentence. There may also be a better word for "statements". -- Jitse Niesen (talk) 13:35, 19 June 2006 (UTC)
 * I like Jitse's sentence &mdash; how about "conjectures" instead of "statements"? &mdash; Paul August &#9742; 14:56, 19 June 2006 (UTC)
 * This sounds good, but the This process may take hundreds of years. part is just hanging there like a quick afterthought. Would it be too much to include it in the previous sentence? With Paul's suggestion we'd have ..
 * &hellip; Mathematicians explore these and related concepts, aiming to formulate conjectures and establish their truth by rigorous deduction from axioms and definitions, a process that may take hundreds of years.
 * My main concern now is that it may give the impression that a new conjecture might take hundreds on years to formulate, again implying that maths is mostly complete. Any idea's how to reword it to avoid that confusion? Cheers -- darkliight [&pi;alk] 06:30, 20 June 2006 (UTC)
 * For the reasons Darklight gives and others I don't think tacking that phrase on the end is a good idea. So to be clear I would support the sentence:
 * ''&hellip; Mathematicians explore these and related concepts, aiming to formulate conjectures and establish their truth by rigorous deduction from axioms and definitions.
 * Like Jitse, I'm not particularly happy with the sentence: "This process may take hundreds of years." The idea that some conjectures take a long time to be proven true or false, is not important enough, in my view, to warrant being in the lead. And while it might be a way to help convey that mathematics is continuing to grow, I don't think it is the best way. Better might be to deal with this more directly with something like:
 * &hellip; Mathematicians continually explore these and related concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.
 * I've also added here "appropriately chosen" to convey the idea that the axioms and definitions are subject to change as well.
 * Paul August &#9742; 15:53, 20 June 2006 (UTC)
 * Agreed about the hundreds of years sentence. I like your version, it gets the point accross and removes the need to for the word results nicely. Good job. Cheers -- darkliight [&pi;alk] 03:45, 21 June 2006 (UTC)

I haven't been around much for a few days, but was thinking of "confirm" instead of "justify/verify". I think the new formulation has removed the need for this solution, but I'll leave it here in case it comes in handy later. Stephen B Streater 20:43, 20 June 2006 (UTC)


 * Paul's latest suggestion looks excellent to me, so I implemented it. -- Jitse Niesen (talk) 04:16, 21 June 2006 (UTC)


 * What point is being made with the word "continually" in "continually explore"? I presume it is not supposed to contrast with "occasionally explore".  Is anything wrong with just "explore"?  McKay 08:01, 21 June 2006 (UTC)

The word "continually" really doesn't work, here. Just "explore" is better.

Unless you want to make it, "to boldly explore where no mathematician has gone before". ; )

Rick Norwood 12:36, 21 June 2006 (UTC)


 * I take Rick and McKay's points. I've removed the word "continually". The point was to emphasize that the development of mathematics is ongoing, to help dispel the common misunderstanding that mathematics is a dead discipline. Perhaps "continue to" would have been better. However I think the lead probably deals with this adequately as it is. Paul August &#9742; 13:48, 21 June 2006 (UTC)

Popper again
Still unsure about this falsafiability. I found the following quote from the paperPopper as a philosopher of mathematics
 * Popper is not usually regarded as a philosopher of mathematics. As mathematical propositions fail to forbid any observable state of affairs, his demarcation criterion clearly divides mathematics from empirical science, and Popper was primarily concerned with empirical science.

which could the the require citation. --Salix alba (talk) 11:54, 28 June 2006 (UTC)


 * This definitely doesn't back up the claim about what Popper himself believed. It may be relevant enough to include in the article, but previous discussions show that some people don't agree with the assertion, so it is only one POV. JPD (talk) 12:03, 28 June 2006 (UTC)


 * To establish falsifiability all that is required is a few clear examples that illustrate why certain mathematical expressions cannot be implicitly trusted as being representative of something real. I submitted such an example this morning but was rv'ed almost immediately by JPD. Archived discussions about this seem to circle around the semantics of Popper's position on Math, rather than tackling the validity of the idea that science (our knowledge of the real world) is a very restricted subset of the realm of mathematics. My rather trivial example, for what it's worth follows: $$

...{n^2\over m^2}+{o^2\over l^2}+{p^2\over k^2}+{q^2\over j^2}+{r^2\over i^2}+{s^2\over h^2}+{t^2\over g^2}+{u^2\over f^2}+{v^2\over e^2}+{w^2\over d^2}+{x^2 \over c^2}+{y^2 \over b^2}+{z^2 \over a^2}=1 $$. Although this formula might satisfy the requirements of mathematical logic and could be differentiated and integated, such a multidimensional construct would be quite impossible in nature. Imre Lakatos, has used similar reasoning in his review of mathematical logic.Geologician 15:51, 28 June 2006 (UTC)


 * I don't think this quite addresses the point. Previous discussions have also featured disagreements concerning whether mathematics satisfies Popper's criteria or not. The question is not whether mathematical expressions can be trusted as being representative of something "real" (by this I assume you mean physical), but what it means to be falsifiable. In this example, it is not clear what it means to say whether the "multidimensional construct" is possible or impossible in nature. JPD (talk) 16:03, 28 June 2006 (UTC)


 * The expression implies the existence of at least 13 dimensions, more than enough for most versions of string theory. Even if there are 26 dimensions the mathematical realm is not limited by that constraint.Geologician 16:41, 28 June 2006 (UTC)


 * If you interpret dimension as degree of freedom, then the physical world has quite a few. Stephen B Streater 18:46, 28 June 2006 (UTC)


 * Constructive idea Stephen. Say there are 12 degrees of freedom [[Image:Degrees of freedom.png]]:
 * sticking with euclidean geometry, these apparently could occupy the same dimension space described in string theory. So math expressions that extrude beyond that realm of reality remain 'Beyond the Pale'. Geologician 09:37, 29 June 2006 (UTC)


 * That diagram is meant to be 12 degrees of freedom of a hydrogen molecule. There are a lot more complicated systems out there than a hydrogen molecule. Your argument seems to rely on a particularly restrictive idea of what can represent something physically real and doesn't at all address other viewpoints. JPD (talk) 11:15, 29 June 2006 (UTC)
 * I am purposely avoiding semantics and retricting my argument to reality. Reality is about expressions like 'If you knock your head aginst a brick wall you will get hurt" "Water is wet" Simple things that human beings have known since the dawn of time, but more recently may have eluded the attention of some academics. Geologician 14:02, 29 June 2006 (UTC)
 * I fail to see how the question of whether maths is a science can be dealt with without semantics. The notion of science is not something I can knock my head on, so we can't restrict ourselves to that sort of "reality". JPD (talk) 16:18, 29 June 2006 (UTC)


 * And you don't need to go to string theory for physically meaningful examples of spaces with 26 or more dimensions. A classical model of the solar system will live in a phase space with at least 9x6=54 dimensions (and more if you want to model satellites, minor planets, comets etc.). Gandalf61 12:25, 29 June 2006 (UTC)
 * Introducing phase space is a red herring. Phase space is merely an aid to _visualization_ of (states of) matter that itself is possibly confined to the restricted dimensions allowed by string theory. Just because something can be visualized doesn't mean its relevant to reality. For example, I can quite easily visualize a moon made of green cheese. Pressure, temperature and composition are not dimensions, simply mathematical constructs that describe the state of matter in space and time. Let's concentrate on attempting to define limits beyond which mathematical expressions cease to be relevant to the real world. Geologician 14:02, 29 June 2006 (UTC)
 * With all due respect, I think it is insisting that expressions are only relevant to the "real world" in terms of your "dimensions", and even discussing relevance to the physical world are the red herrings here. See Salix alba's comment below. JPD (talk) 14:31, 29 June 2006 (UTC)


 * Geologician - if you read beyond the introductory paragraph in the phase space article you will see that phase space is not the same as a phase diagram in physical chemistry. The concept of phase space has nothing to do with "visualization"; it is not concerned with states of matter; it has not connection at all with string theory, which it pre-dates by over a century; and the co-ordinates in phase space are positions and momenta, not pressure, temperature and composition. And it is intensely relevant to answering questions such as "will Halley's comet impact the Earth in 2061 ?", which seems like a "real world" problem to me. Gandalf61 15:50, 29 June 2006 (UTC)
 * Math is all about visualization. Position and momentum are simple functions of mass, time and distance so they fit neatly within the constraints of string theory dimensions. (Direction is not a dimension) Geologician 23:27, 29 June 2006 (UTC)


 * Hmmm. Well, either you really have a really flawed understanding of the basic terms and principles of mathematics and physics, or you are pretending so in order to provoke a response. In either case, I see no point in continuing this discussion. Gandalf61 07:51, 30 June 2006 (UTC)


 * Care is needed to distinguish mathematics from mathematical physics. The latter attempts to address science (our knowledge of the real world)(Geologician above) whereas Pure mathematics does not - Hardy's apology being a prime example of the lack of real worldness.
 * For lack of real-worldness we don't need to consider anything as exotic as higher dimensional spaces, some of the most basic abstractions of mathematics: say the concept of a surface do not correspond to anything physical, as no physical entity will have zero thickness. There is a good section in one of Feynman's books on distinction between abstract mathematical entities and real world entities. Whether that most basic concept of the real numbers is more than a useful abstraction is a good question: is space infinitely divisible? Is this statement falsifiable?
 * In any case we still have
 * Karl Popper believed that mathematics was not experimentally falsifiable and thus not a science.
 * without any direct evidence that he was so crude in his thinking. A better statement might be
 * Some authors have used Poppers's concept of experimental falsifiability to distinguish mathematics from the empirical sciences. Ref Glas
 * --Salix alba (talk) 13:33, 29 June 2006 (UTC)
 * That sounds much better to me, although it probably would be less "weaselly" to mention some of these authors in the text, as well as giving a reference. JPD (talk) 13:52, 29 June 2006 (UTC)
 * In response to JPD's 14:51 suggestion re. Salix alba's comment above. I am well aware of the distinction between Pure and Applied Math. However surely Popper's concept of experimental falsifiability is relevant only to the latter. Applied math (or mathematical physics) seeks to use the most appropriate tools available in the mathematical grab-bag to describe physical phenomena. These phenomena are prepared for mathematical analysis by making measurements. Measurements convert the observation into real numbers curtailed to a discrete number of significant digits. It is at that point that Salix alba's concept of infinitely divisible space breaks down. Numbers only become relevant to the real world when something can be measured. And no result of a calculation can attain more precision than the component measurement made with the least precision. Regrettably no scientific calculator software facilitates the tracking of the least significant digit. My point is that mathematics tend to promote the idea that it is "the Queen of the Sciences" but fails to deliver when called to account. Geologician 15:12, 29 June 2006 (UTC)
 * The article makes it quite clear that the use of the word "Sciences" in that description is not the meaning most often used today. We are talking about mathematics, not mathematics as used in physics. Your claim that Popper's concept of falsifiability is not relevant to mathematics in general is exactly what is disputed. We can't put examples into the article in a way that assumes your POV. I probably agree with you more than it seems, but I think it is important for the article to be in NPOV terms, using wording such as that which Salix alba suggests. JPD (talk) 16:18, 29 June 2006 (UTC)
 * Popper used the word in the modern sense so we shouldn't be distracted when we consider Salix alba's proposition that: Karl Popper believed that mathematics was not experimentally falsifiable and thus not a science. Apparently no-one has yet come up with a citation to support this, however 'falsifiability' derives from statements like "no empirical hypothesis, proposition, or theory can be considered scientific if it does not admit the possibility of a contrary case."  As mathematics is a collection of these components, one cannot test the entire collection by Popper's criterion, one has to consider each component individually. Otherwise the whole of mathematics stands or falls by the reliability of one of its components. Therefore it is perfectly legitimate to point out that statements like "the area of a circle of radius 1.7 cms is 9.079 sq.cms" is in significant error and the true area is actually 9.1 sq cms. Consequently the key proposition that the area of a circle is pi times the radius squared is also wrong, if it lacks the important caveat that the result cannot have more significant figures than the radius. And so we can proceed with a hatchet though the entire corpus of mathematical formulae insofaras they relate to the real world. QED Geologician 20:53, 29 June 2006 (UTC)
 * In mainstream mathematics the hypotheses, propositions and theories are not empirical. While it is possible to do "experimental mathematics", in which conjectures may well be falsified, the claim that in Euclidean geometry the area of a circle with radius r equals πr2 has an entirely different status. Should the measurements of a physical embodiment of a circle reveal something different, three possible explanations are: (1) our measuring apparatus is defective or insufficiently precise; (2) we are measuring an imperfect physical model of the ideal circle; (3) the local geometry is not Euclidean. The experiment will not put the claim in doubt. It is increasingly unclear what you are trying to say. --Lambiam Talk 21:53, 29 June 2006 (UTC)
 * P.S. A quote from Einstein (according to Wikiquote): As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. (I'd like to add that as far as anything refers to reality it is not certain.) --Lambiam Talk 21:58, 29 June 2006 (UTC)
 * You'd like to, but can't be entirely certain ;-) Stephen B Streater 08:24, 30 June 2006 (UTC)