I chose 'other' because I don't think there is any choice for either 'all of the above', or 'none of the above'. I think mathematics is a science and a language. Therefore I would think of it more as a series of discoveries which have evolved over the centuries. Arithmetic is simply the natural outcome of the need to count things with a natural progression to various simplifications for doing so. I believe the Arabs invented Algebra as a kind of grammar of the 'language' of mathematics. Geometry is the result of observation of the relationships between natural phenomena, the ratio of a circle to its diameter, the perpendicular to the horizontal. The astronomical observations of Kepler are what probably led Newton and Leibnitz (independently) to discover calculus. The fact that two mathematicians independently derived a similar discipline using different notation would seem to suggest that, like all of the physical sciences, over time, discovery was inevitable. I guess I would call mathematics a science for the quantification of natural laws. I confess that I have no idea of whether any mathematician, philologist, or linguist would agree with anything I said here, but, what the heck, its a free country. Maybe I just don't understand the question. +++++++++++++++++++++ 04-24-05, 10:52 PM jusork If you have a single apple and another single apple on the ground, you have two apples on the ground. If two more fall from the tree, you have two more apples. Since you have two apples and another two apples, you'll always have four apples total. Math is a discovery. Actual numbers though are a representation and they are an invention. Even without numbers, you still have apple, apple, apple, apple which, put together, make a unique group of apples. Math may be eternal, but it's not apart from the universe. It's not like it transcends into some divine reality or something. It's a part of this universe. (There is after all only one universe, right? If there are multiple universes, then math would be an eternal reality of all universes.) And it doesn't take a mind for a concept to exist. Individual apples come together in certain forms whether they are concieved or not. The standards of math are there just the same. If you've got 149 apples in one giant bucket, 149 apples are still in one bucket. And 23 of them are still green and 55 of them have brown spots whether they realize it or not. And the degree of a brown spot exists in nature so if the brown is at a certain brownness, then it will be apart of that browness in nature.
04-24-05, 11:42 PM Professor This is a fascinating question that has been the subject of much debate by philosophers. For instance, there's a deep discussion of this in Pi in the Sky by John Barrow, though I'm not sure he comes to any particular conclusion.
Obviously our notation, and even our conceptual organization of mathematical knowledge, is highly culture-bound.
But on a more abstract level, I personally favor the notion that mathematical truths are universal and "out there," hence discovered rather than invented. In other words, I favor the view that intelligent aliens in a distant galaxy would still agree with us on the truth of 2+2=4, the Pythagorean Theorem, Fermat's Theorem, etc. It's hard for me to imagine otherwise. Yet many would disagree.
However, the process of mathematical discovery is perhaps more akin to invention, requiring creativity and construction, different in charcter from scientific discovery.
04-25-05, 12:05 AM methos I agree with everything frankvan said except for his classification of math as a science. Of course, many mathematicians would agree with him rather than me.
There are certainly similarities in technique and mindset, but there is a fundamental difference. Science applies to physical reality. When it conflicts with reality, it is wrong. A theory which is flawless on paper is wrong if an experiment disproves it.
Math doesn't suffer such a constraint, and this sometimes lets math advance ahead of science (String theory uses math conceived at least a century before there was any need for it), but sometimes advances math in directions that, although mathematically "true," are meaningless to our physical reality. To me, this differentiates it from science.
The example of string theory is an interesting one, though. It has been criticised as not really being science (for essentially the same reason I don't consider math to be science). Although it is mathematically "elegant" (as Brian Greene describes it), it has yet to produce results testible by experiment. Since it is not falsifiable, some of its opponents say, it is not really science.
I agree that it is probably a discovery, not an invention, mostly because so many people have come up with various elements of mathmatical rules independently. I've been thinking a lot about this, especially because I'm currently reading a great book called The Mind of God that deals, in part, with this very issue.
Anyway, this idea does bring up some strange questions. For example, let's take the Pythagorean Theorem. A squared plus B squared equals c squared on a triangle seems to be a discovery, not an invention, since it is true of all triangles.
Does this mean that it is *necessary* that the pythagorean theorem be true? Does this mean that it couldn't have been possible for there to have been a universe in which every triangle followed a different rule? Or a universe in which triangles can't or don't exist, even in theory? To me, I can't possibly imagine this, so it seems to me that such a rule is *necessary.*
The next thing this brings me to is the question of whether there are possible shapes that we have never seen. I imagine there must be, for example in a 5-dimensional universe or something like that. Mathmaticians have discovered such shapes, haven't they?
This is sort of hard for me to get my head around, so excuse me if this is unclear. If the mathematical rules, like the Pythagorean Theorum, are eternally true (necessarily true), then where are they? My first instinct is to say that they are in the things. For example, the Pythagorean theorum exists nowhere except in triangles themselves. However, if there are other shapes that we have never seen but can understand abstractly through math, you don't really need the triangle itself to have the Pythagorean theorem. So if math is a discovery, and if it does not have to reside in shapes around us, where is the thing being discovered?
04-25-05, 07:22 PM frankvan
quote: A squared plus B squared equals c squared on a triangle seems to be a discovery, not an invention, since it is true of all triangles.
I'm sure you know better, but it is NOT true of all triangles. Only right triangles.
04-25-05, 09:05 PM Sarai Yes, I do know better (or at least DID, when I was in high school!). Big Grin Red Face
Thanks for the correction, though. My general question is still the same, despite the slip-up... if the question makes any sense to anyone!
04-25-05, 09:41 PM jusork
quote: Originally posted by Sarai: This is sort of hard for me to get my head around, so excuse me if this is unclear. If the mathematical rules, like the Pythagorean Theorum, are eternally true (necessarily true), then where are they? My first instinct is to say that they are in the things. For example, the Pythagorean theorum exists nowhere except in triangles themselves. However, if there are other shapes that we have never seen but can understand abstractly through math, you don't really need the triangle itself to have the Pythagorean theorem. So if math is a discovery, and if it does not have to reside in shapes around us, where is the thing being discovered?
I wondering if we could better answer this if we think about where logic comes from. At least I think that would answer it. It seems to be along the same lines. Would logic also not be able to be found in things themselves? Is A = B and B = C, then is A = C a good example? Someone back me up.
04-26-05, 09:02 AM frankvan . The original question, if I understand it, is: "Is math a discovery or an invention?". And the discussion seems to be whether or not the right triangle contained the Pythagorean theorem before there was a Pythagorean theorem? And if it did not, the theorem would be an invention, if it did, the theorem was a discovery. There are, of course, other relationships between the three sides of a right triangle besides that particular one, so that at some earlier or later date some other thinker could 'invent' or 'discover' that the sum of the three angles formed by the sides totaled 180 degrees. I believe that the triangle itself contained all of the relationships waiting to be "discovered" and Pythagoras came along, discovered one of them, and invented a means of demonstrating that relationship. Q.E.D (Quite Easily Done, as a math instructor long ago was fond of saying.) And, I'm sure that jusork is right, Pythagoras probably used logic to arrive at his theorem, but I don't know how or why he thought that beans were evil. Confused
04-26-05, 11:01 AM Sarai The answer "It's a discovery" brings me to the second question, which is what exactly is being discovered, if math exists in theory, and doesn't need the "things" in order to be true (as is the case with mathmatics dealing with shapes in other dimensions). What do you think?
04-26-05, 12:16 PM methos The mathematics of shapes is an interesting example for you to chose. Euclidean geometry in 3-dimensional (or, in calculations often only one or two-dimensional space) once seemed enough to describe the way things really were.
Einstein fused time into the picture as a fourth dimension in his special theory of relativity (100 years ago this year)
Then, when formulating his general theory of relativity about 2 decades later, Einstein realized that Euclidean geometry in any number of dimensions was an incomplete picture of the world. For that, he needed a new form (Reimann geometry). That geometry, where space is curved, was formulated without seeming to apply to our reality nearly 100 years before Michael Grossman, a mathematician taught it to Einstein who showed that it does apply to the real world after all.
Now, for string theory (the proposed "theory of everything"), it turns out that a strange 12-dimensional space, a space that already existed in the minds of mathematicians, may apply to our world after all. So, after all my blabbering... yes, mathematicians have worked out the geometries of spoace with more than three dimensions and theoretical physicists are now hypothesizing that they may be more than mathematical abstractions.
04-26-05, 12:48 PM Sarai Methos: Beyond the fact that it works mathmatically, what reason do scientists have to believe in string theory?
04-26-05, 02:18 PM methos Ok, I went through a lot of blabbering at the bottom which may or may not be completely coherent, so I've said it a different (hopefully more straightforward) way here at the top.
The simple answer is string theory has not produced any prediction that, if verified experimentally, would say that it is right and older theories are wrong (or, more accurately, incomplete). It's attraction really, in my opinion, lies in this idea of elegance. To a scientist, describing more with less is elegance (perhaps an overgeneralization).
(1)Quantum mechanics certainly seems complicated, but in it we combined two separate sets of laws (one for waves and one for particles) so that everything can be described with the same equations. Similarly, string theory hopes to combine relativity and quantum mechanics.
(2)A hundred types of atoms, one having fundamentally 'goldness' and another 'silverness', etc. is more complicated than all atoms being composed of different combinations of only three types of particles. The latter is perhaps more difficult to learn, but it is a simpler idea in that it doesn't use 100 separate sets of laws but only one. String theory seems to show that there is in fact only one thing in the universe, a string, and all the smallest particles are just this string vibrating in different ways.
So, string theory is controversial and there are many who believe the attention and alottment of resources to it is unfounded. No one, not even its staunchest supporters, would say that it is anywhere close to proven
----------------------------------------
String theory offers a certain "elegance" in that it seeks to explain the entirety of reality through a single process (the vibration of "strings"). Modern science has been pushing in this direction since Newton (at least). He showed that the heavens and the earth follwed the same rules, his laws of motion. Maxwell combined light, electricity, and magnetism. Einstein combined time and space, matter and energy, and acceleration and motion. De Broglie, Einstein, Bose, Shrodinger and others combined particles and waves. Various scientists showed that all the different elements are made up of the same things. Gell-Mann showed that many of the multitudes of fundamental particles are not really so fundamental after all but are instead made up of fewer small particles in different combinations. Others such as Feynmann gave us rules for transforming these particles into one another and showed the interconnectivity of the other forces (strong and weak) with electricity and magnetism.
The next challenge is the fusion of gravity with the other forces - gravity is described by relativity and the other forces with quantum mechanics (hence, it is sometimes called the search for a quantum theory of gravity). This was Einstein's goal because he saw the world as following a single set of rules and that set of rules as knowable. when Einstein started his quest, there was no obvious disagreement between the two theories, it was simply that they described separate phenomena. Now, we have examples (black holes being the most often cited) of direct conflict between the two sets of rules. If you try to combine the theories, everything breaks down.
String theory is more of a promise than an actual solution. Relativity needs space to be warped smoothly. Quantum mechanics says that everything, including space, is chaotic at small scales. String theory smooths some of this quantum roughness to create something that, while it can't really said to be correct (yet), at least does not break down.
The problem is that, for all its "elegance", string theory hasn't produced anything truly testable. Einstein's relativity was elegant, but it was hotly debated up until proof started to appear. For example, Einstein predicted that based on his theory, gravity should bend light by a certain amount while, based on Newton's theory, it should bend only half of that amount. Some British scientists (whose names escape me) went to Antartica during an eclipse to experimentally see who was right and came down on Einstein's side. So far, string theory hasn't proiduced anything like that.
04-26-05, 07:03 PM aminator2002 The nice thing about math is that it is truth.
That's about as eternal as it gets... if we disappear tomorrow, math will still be here, and its principles have always been here whether humans knew it or not.
It is a discovery of truth by way of a system that grows over time with further human understanding.
04-30-05, 12:18 PM Sarai I'm not sure I agree with you, aminator, that math is truth. In another thread, we've just had a long discussion about one particular mathmatical statement that I think shows that even when the math is done correctly, it can describe reality in ways that are completely wrong. If math were 100% truth, you and I would not be having this conversation, because movement and the advancement of time would be impossible (see the link for more information).
I chose "I think it is a discovery and math is an eternal reality" on the poll, but if I could, I would change my vote. Now, I would choose "I think math is an invention that humans use to understand the world." And, in fact, I would add the world "fallible" before the word "invention."
04-30-05, 12:34 PM Sarai
quote: Or maybe math is fallible in a finite world, but not fallible in itself, which doesn't exist physically in this world, but exists in a non-finite, eternal, timeless logic. And logic can also be applied to the nature of the universe. The math that shows itself in nature is only a reflection of the truth. 1 + 1 exists in the math place of truth and, hey!, that piece of logic in math is also logical in the finite world. Math is a truth, but it's not the same thing as reality. Math goes by math logic which comes in both infinite and finite. The universe goes by finite logic, which sometimes uses the finite math logic. It just describes reality. Math isn't an element in reality, but reality is an element in math. Is there anything in reality that math can't describe?
So yeah, I guess it would be math describing reality incorrectly...because math doesn't exist hand in hand with reality. Math is still a truth. Remember, everything doesn't revolve around the universe.
I'm not a mathmatician so I could be missing something, but it seems to make sense.
Hey, Frank!
This is a quote from Jusork in this thread. I brought it over here because I think it belongs, more properly, under this question.
Jusork, I think I see what you're saying. But if it were true that math exists outside somewhere (that is, outside of consciousness and outside of the things it is supposed to describe), where does it exist? A universe that had no things and no consciousness would not have math, right? Without the things math is supposed to describe, and without the consciousness to use it, math is dead. Therefore, when math gets into theoretical realms (by this I mean realms that we cannot experience to test to see if it is really describing things), it is just a game, not truth.
04-30-05, 02:36 PM frankvan QUOTE: "As far as I can think of, there are two types of failures of math to describe reality.
One is where the mathematical description is incomplete.
In this case, Zeno considers the movement in space without the movement in time. Sarai's argument similarly takes the movement of time without considering the movement through space. It is only when you take them simultaneously that it works (it doesn't take infinite time to move a finite distance, even by the arbitrary half-step method, if you consider that the change in time is gettign ever smaller with each step as well)." Methos.
I think I'm beginning to see part of the problem with this discussion. Both of you, Sarai and Methos' appear to be saying that what Zeno's paradox states is an example of mathematic's failure to represent reality. Well, I wouldn't call that - what Zeno does - is applying math. Failing to take time into consideration, or failing to take distance into consideration when we are talking about motion is certainly a not a credible example of mathematics. If that is what Sarai is calling mathematics. And while we're at it, I wouldn't agree that Newton's laws of motions' failure to endure later discoveries, in any way discredits its use in the reality with which it was contemporary, or indeed with most earthbound calculations to the present time.
Come to think about it, (the original question), What really is the difference between a discovery and an invention? Confused
04-30-05, 05:55 PM Sarai
quote:
I think I'm beginning to see part of the problem with this discussion. Both of you, Sarai and Methos' appear to be saying that what Zeno's paradox states is an example of mathematic's failure to represent reality. Well, I wouldn't call that - what Zeno does - is applying math. Failing to take time into consideration, or failing to take distance into consideration when we are talking about motion is certainly a not a credible example of mathematics.
"You must walk a mile. How many feet will you cross?"
That is a legitimate elementary school question for a math exam. It does not take time into consideration. It is simply asking how many feet are in a mile.
"You must walk a mile. How many times can the distance you walk be divided by two?"
This is also a legimate mathmatical question. It is simply asking how many divisions can be made in a mile. The answer is an infinite number of divisions. You do not need to take time or speed into consideration to see that a mile is divided into an infinite number of parts.
quote: Come to think about it, (the original question), What really is the difference between scovery and an invention? Confused
A discovery is something you find that already exists. An invention is something you create that did not exist before. Electricity is a discovery; the ability to harness electricity in a usable form (such as television) is an invention.
04-30-05, 07:25 PM frankvan
quote: That is a legitimate elementary school question for a math exam. It does not take time into consideration. It is simply asking how many feet are in a mile.
There are many math problems and exam questions that DO NOT take time into consideration. But when you say that dividing that mile into an infinite number of imaginary increments results in the consumption of infinite time -so that you can "never reach your destination" - you are taking time into the equation, whether or not you realize it.
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Perhaps this explanation is simpler and easier to understand than my poor attempts:
Now the resolution to Zeno's Paradox is easy. Obviously, it will take me some fixed time to cross half the distance to the other side of the room, say 2 seconds. How long will it take to cross half the remaining distance? Half as long – only 1 second. Covering half of the remaining distance (an eighth of the total) will take only half a second. And so one. And once I have covered all the infinitely many sub-distances and added up all the time it took to traverse them? Only 4 seconds, and here I am, on the other side of the room after all. And poor old Achilles would have won his race. from: http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp +++++++++++++++++++++ 04-30-05, 09:01 PM jusork
quote: Originally posted by Sarai: Jusork, I think I see what you're saying. But if it were true that math exists outside somewhere (that is, outside of consciousness and outside of the things it is supposed to describe), where does it exist? A universe that had no things and no consciousness would not have math, right? Without the things math is supposed to describe, and without the consciousness to use it, math is dead. Therefore, when math gets into theoretical realms (by this I mean realms that we cannot experience to test to see if it is really describing things), it is just a game, not truth.
It exists as a concept. Not a concept just in people's minds, but an eternal concept, a truth, a truth that is sometimes also a truth in reality. Where 2 apples and 2 other apples equal four apples, 2 and 2 also equals four in the non-physcisal logic. 2 of anything and 2 of anything equals four even outside of those things themselves. Just like Ami said, even if all minds were gone, 2 + 2 would always equal 4. It's becasue that logic exists as a truth, a truth not just in objects, but also in the eternity of logic. I don't think the truth of math relies on the reality that holds some of math. If the universe ended, 2 + 2 would still equal 4. It just wouldn't have anything in the world to show that.
Can't math be a truth that is used as a tool?
04-30-05, 11:08 PM Sarai
quote: Originally posted by frankvan: Perhaps this explanation is simpler and easier to understand than my poor attempts:
Now the resolution to Zeno's Paradox is easy. Obviously, it will take me some fixed time to cross half the distance to the other side of the room, say 2 seconds. How long will it take to cross half the remaining distance? Half as long – only 1 second. Covering half of the remaining distance (an eighth of the total) will take only half a second. And so one. And once I have covered all the infinitely many sub-distances and added up all the time it took to traverse them? Only 4 seconds, and here I am, on the other side of the room after all. And poor old Achilles would have won his race.
Well, you're right - that final explanation, finally, does make sense to me! Sorry you had to go through such gymnastics to get it through my thick head, but in the end, you have finally explained it to me, Frank! I'm not quite sure why, but I'm relieved. Thanks.
Jusork and Aminator, Frank has just convinced me that you're both right as well. Big Grin
05-01-05, 11:54 AM frankvan Well, I am also relieved that we are all freed to cross the room, jump our frogs, or race tortoises once again. Cool
05-01-05, 01:33 PM jusork Wow! Haha. Frank is a miracleworker! Big Grin
05-01-05, 06:57 PM frankvan
quote: Originally posted by jusork: Wow! Haha. Frank is a miracleworker! Big Grin
In future, I prefer to be called "Your Eminence". Wink
05-01-05, 09:26 PM jusork Haha. Wink
05-01-05, 10:36 PM Sarai
quote: Originally posted by frankvan:
quote: Originally posted by jusork: Wow! Haha. Frank is a miracleworker! Big Grin
In future, I prefer to be called "Your Eminence". Wink
I'll call you anything you like, Frank, if you'll riddle me this.
Sorry to ask so many questions, but you see, I have an infinite number of them and only a finite amount of time to get the answers. Wink
05-02-05, 08:27 AM frankvan Wow, my wife is coming with the groceries and I'm expected to assist. But I also have to get into this thing before someone else beats me to it. A quick hunch. The problem is different than the achilles one.In this case there seem to be two infinite converging sequences the "on" one has the limit as the denominator azpproaches infinity of 2 minutes and the "off" one approaches 1 minute. But the one minute sequence is up befor the "on" one so the light is on at the end of two minutes. Now just watch someone jump in here before I get the groceries unloaded and prove me wrong. As Arnold says, "Ahl be baaaak".
05-02-05, 12:21 PM frankvan Okay, I retract the previous answer. There is only one series and the total time is 2 minutes.It must be obvious first of all that we cannot have an infinite number of anything, we can only have the concept of a direction without any restraints. We determine the sum of a converging sequence of numbers by imagining that no matter how far we carry the summation it can never go higher than a certain limit. Thus a series of fractions of the form 1/n as n goes toward infinity, which starts at 1 will end at 2, because the fraction starts with a value of one and adds fractions whose value varies inversely with the growth of the 'n' term.
Consequently, the lamp which is turned on for one minute and then off for one half minute, then on for 1/4 minute, off for 1/8, etc will be on and off alternately for a total of 2 minutes. At the two minute mark, I don't think we can state with any certainty, whether the lamp will be on or off. for the reason that we have no way of knowing how many minuscule intervals will result from dividing up a two minute total. It must be obvious that there would be more intervals in a three minute limit than in the two minute one. Therefore the last 'on' or 'off' would have to be greater than the last 'on' or 'off' in a three minute sequence, so the next to the last division would have to be something greater than zero. We can't posssibly state that there are an infinity of intervals between the numbers 'one' and 'two' and also that there must be two times infinity between 'one' and 'three'. So, my second answer would be that just as we can't determine whether the lamp will be on or off when starting in the on state, I believe the same answer applies if it starts in the 'off' condition.
And I'm sticking with that answer until someone contradicts it. Please, Sarai, remember I'm an old man, losing brain cells at an alarming rate. I could hurt myself!! Smile
05-02-05, 01:57 PM Sarai It appears that you're right - this is an impossible question to answer. Thanks anyway, Your Eminence.
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