Mathematics - Invented or Discovered

[Royce]

I've been reading Roger Penrose's " The Emperor's New Mind." He brought up a question that I once thought I knew the answer. "Is Mathematics invented or discovered?" I.E. Is Mathematics a purely mental construct or does(do) Mathematics exist, as in Plato's forms, somewhere, as Truths of Reality that we discover rather than invent?

And, by extension is Logic invented or discovered truths of the universe?

Think about it. Then give us your thoughts on the matter. It is not really as simple or straight forward as it first appears.

[selfAdjoint]

I have posted my beliefs before, but I'll recap here. I believe mathematical objects and relationships are all ideas in human minds. Pairs of objects exist in nature, but the number two is a human idea. Likewise objects may exist in three different places, but a triangle does not exist until a human notices it.

These mathematical ideas have a property called being well-defined. That means they can be completely understood, and people equally informed about them cannot disagree about their nature. There can be vigorous disagreements about metamathematical subjects - constructivism, formalism, whatever. But not about the nature of the mathematical ideas, This well-defined character is the reason they can be mistaken for things outside the mind.

[Royce]

That was my thoughts on the subject until I read Penrose's book. He brought up such things as Mandelbrot's Set or series, fractals, as an example of something accidentally discovered rather than constructed within our or his mind with this intent. Below is a link to just once site with the image of the set used. http://mathworld.wolfram.com/MandelbrotSet.html. This is just one example that he used.

His point and my question is do we really purely invent Math, Number Theory, Algebra, Set Theory, Calculus etc. or does the study physical reality and the world around us lead us to discover it to better describe and understand the universe as it is?
There are a lot of things in Math that were not invented knowing what the results and uses would be but found to be previously unknown and undreamed of products of existing Math. Another example of discovery is the Great Attractor in Chaos Theory.

[Stevo]

Read Wittgenstein's "Tractatus".

[selfAdjoint]

That was my thoughts on the subject until I read Penrose's book. He brought up such things as Mandelbrot's Set or series, fractals, as an example of something accidentally discovered rather than constructed within our or his mind with this intent. Below is a link to just once site with the image of the set used. http://mathworld.wolfram.com/MandelbrotSet.html. This is just one example that he used.

His point and my question is do we really purely invent Math, Number Theory, Algebra, Set Theory, Calculus etc. or does the study physical reality and the world around us lead us to discover it to better describe and understand the universe as it is?
There are a lot of things in Math that were not invented knowing what the results and uses would be but found to be previously unknown and undreamed of products of existing Math. Another example of discovery is the Great Attractor in Chaos Theory.

I know that Penrose is a Platonist and believes that mathematical structures exist in some "other reality" and are discovered rather than imagined.

But I don't see a true dichotomy between imagined and discovered. The Mandlebrot set is derived from the inverse of a simple complex number function. Its boundary separates the region of the complex plane where the inverse converges from the region where it does not converge. Now in my opinion that derivation is entirely a mental operation. It "looks" like discovery to the mathematician, but that is as I said before because of the sharp well-defined character that the function has. You work it out and plot the results (as Poincare did initially). Or to get more detail you program the steps you have worked out (mentally) on a computer and generate one of those pretty false color images. None of this requires us to imagine the set existing outside us in some world of Platonic forms.

You can teach another person how to generate the Mandlebrot set and then, having the necessary ideas in their heads, they can do it too.

[MaxNumbers]

This question seems to rest mainly in the definition of mathematics. If we say that mathematics is a language to describe reality, then I think it makes it all much clearer. For example, we see pairs of things in nature, so we give it the term "two" and begin to form the idea of "two" separate from nature. This is exactly what we do in language-- we know of emotions, and we give them names that begin to encompass the essence of anger, joy, fear, etc.
If mathematics is a linguistic tool, then we invent the terms, the names, but we are simply representing nature.
But, maybe this is just a cheap way out of the dilemma.

[sol2]

Self Adjoint,

I remember these previous discussions and how they arose :smile:


LAKOFF: This view was not empirically based, having arisen from an apriori philosophy. Nonetheless it got the field started. What was good about it was that it was precise. What was disastrous about it was that it had a hidden philosophical worldview that mascaraded as a scientific result. And if you accepted that philosophical position, all results inconsistent with that philosophy could only be seen as nonsense. To researchers trained in that tradition, cognitive science was the study of mind within that apriori philosophical position. The first generation of cognitive scientists was trained to think that way, and many textbooks still portray cognitive science in that way. Thus, first generation cognitive science is not distinct from philosophy; it comes with an apriori philosophical worldview that places substantive constraints on what a "mind" can be. Here are some of those constraints:

Concepts must be literal. If reasoning is to be characterized in terms of traditional formal logic, there can be no such thing as a metaphorical concept and no such thing as metaphorical thought.

Concepts and reasoning with concepts must be distinct from mental imagery, since imagery uses the mechanisms of vision and cannot be characterized as being the manipulation of meaningless formal symbols.

http://www.edge.org/3rd_culture/lakoff/lakoff_p3.html

[Janitor]

The threads having to do with this topic are among my favorite at the Physics Forum. I myself vacillate on what the answer is. I can see points on both side of the issue.

[Kerrie]

i say invented only because we do not have another intelligent form of life to prove they also have "discovered" math. math is necessary also for modern humanity, and as the saying goes: "Necessity is the mother of invention"

[Royce]

I have never been able to accept Plato's forms as literally existing somewhere in reality and I don't think that mathematics do either. What I do think, at least in reference to Penrose's ideas is that, along somewhat with MaxNumbers, is that mathematics may be an intrinsic and necessary part of nature, the universe or reality, whichever you prefer. As such we may thing that we invented math but it was by studying, observing or working with nature that we out of necessity discovered it.

I can just see in my mind two cavemen each with an apple and grunting their sound for one then putting them together and Eureka! they "discovered that they had two grunts instead of one thus coming to the astounding and brilliant conclusion that one grunt and one grunt make two grunts. Thus was mathematics born; and, as they say, the rest was history.

I cannot see however some reclusive caveman sitting in his cave and rather than contemplating where and what he was going to hunt, inventing our number system and theory.

This is why I question that mathematics was invented and believe that it was discovered as a basic and intrinsic feature of our reality.

[selfAdjoint]

We appear to have simple counting and arithmetic up to five or seven hardwired in our brains. This is not remarkable, many nonhuman animals do too.

[sol2]

How many animals do you know that can think of another reality under stringtheory :smile:

Platonic considerations are always a interesting feature, as it reveals, human kinds fondness of describing structures. So their was this battle for mankind to debate between discrete and continuity:)

I think royce, you might like the references to this "position" in terms of ideas (http://www.physicsforums.com/showthread.php?goto=newpost&t=24182) and their manifestation?

We have to come out of this cave thing. Heisenberg was fascinated with it? :smile:

And now, I said, let me show in a figure how far our nature is enlightened or unenlightened: --Behold! human beings living in a underground den, which has a mouth open towards the light and reaching all along the den; here they have been from their childhood, and have their legs and necks chained so that they cannot move, and can only see before them, being prevented by the chains from turning round their heads. Above and behind them a fire is blazing at a distance, and between the fire and the prisoners there is a raised way; and you will see, if you look, a low wall built along the way, like the screen which marionette players have in front of them, over which they show the puppets

http://www.ship.edu/~cgboeree/platoscave.html


But sure, there is more to it. Cognitive Science of Mathematics (http://www.superstringtheory.com/forum/metaboard/messages18/345.html)

Some modern theories in the philosophy of mathematics deny the existence of foundations in the original sense. Some theories tend to focus on mathematical practice, and aim to describe and analyze the actual working of mathematicians as a social group. Others try to create a cognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the 'real world'. These theories would propose to find foundations only in human thought, not in any 'objective' outside construct. The matter remains controversial.

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

[Rader]

This is why I question that mathematics was invented and believe that it was discovered as a basic and intrinsic feature of our reality.

Can the infinite fractal curvature of space-time, caused by gravity, be invented, without the already existing mathematics? Certain innate constructs seem to me, to be, a necessity for our reality to unfold. :confused:

[selfAdjoint]

Aren't you talking about physics rather than mathematics? Physics can show how a given facet of nature is well described by a particular piece of mathematics. But it can never be proved (according to what a proof would mean in mathematics) that mathematics exactly describes all the properties of the world.

[Royce]

But it can never be proved (according to what a proof would mean in mathematics) that mathematics exactly describes all the properties of the world.

This is a good point. It is, however, also true that there are many things in mathematics that cannot be exactly described, computed, by mathematics. Much of mathematics are intuitive and based on assumptions.
Euclidean geometry is the first example that comes to mind. Rational and complex numbers are two others.

As Penrose pointed out mathematics itself in incomplete and cannot be completely proven according to Gödel's Theorem regardless of whether it is invented or discovered.

Speaking of Gödel's Theorem, how about that, was it discovered or invented?
Did Gödel know what he was doing before hand? Did he set out to formally prove what he already knew and already knew how to do it as is pretty much a requirement for invention. Very few things are invented accidently.

[matt grime]

What ec=vidence is there to back up the assertion that penrose is a platonist? i don't know, nut sincerely doubt any practisign mathematician is such a person, as anyone who understands the continuum hypothesis, say, will think too.

[selfAdjoint]

I apologize, it was Stephen Hawking who said Penrose must be a Platonist (the two men were close colleagues for many years). This was in a debate about "The Emperor's New Mind". Penrose responded that he regarded himself as a realist.

I believe this is a joke and that Penrose was using the term realist in its medieval sense; one who believes universals are real. Since mathematical axioms are universals, this reduces to being a Platonist in the case of mathematics, but possibly not in any other case.

[Royce]

In "The Emperor's New Mind" Penrose said that he was leaning toward the Platonic view himself in the same sense, I think that I am; i.e. Mathematics is an inherent part of the universe and we do not so much invent it as discover it for some of the most basic theorem's of the ancient Greeks are still true today and seem to be universal truths of and by themselves and not simple "mental constructs" of our minds. I have been paraphrasing much of what he said in the book and am using some of his examples.

I am not a mathematician so I hope that I am being clear enough and not mangling Penrose's ideas and writing to badly. As I said, before reading this book I more or less assumed that Mathematics and Logic were pure abstract constructs of our minds and so invented. I must admit that I am continuously amazed at how accurately these mental constructs can and do describe the physical world around us.

Can this be just a coincidence or is the universe really based on universal mathematic truths that we are still discovering?
It seems that the deeper we delve and the more complex and esoteric that Math and Logic become the more it seems to be discovery rather than invention.

We must remember that invention is a planned event with foreknowledge of what in required and what the desired result is with the stated intend of doing so. Discovery is the use of known facts to find new unknown facts and is often accidental, surprising, often creates more questions than it answers and is often serendipitous and mental leaps.

I must also admit that I have long leaned toward much of Plato's thinking and philosophy rather than that of Aristotle's but have never thought much of forms unless I was taking that idea far to literally myself.

[Janitor]

From the reading I have done on the classification of finite simple groups, there was historically a strong feeling of discovery. Some group theorists were able to show that if there was another sporadic finite simple group bigger than the ones already catalogued, it could only have an order (i.e. the number of elements within the group) of five gazillion two hundred and twenty nine udzillion and thirteen elements. Then with some additional work, they would find--lo and behold!--that there really did exist a group with precisely that order.

[honestrosewater]

How has information theory been applied to the question, if at all?

For example, Shannon's simple information/communication model:

message source -> encoder -> (noise ->) channel -> decoder -> message receiver

Message receiver- us
Decoder- intelligence
Channel & Noise- real world
Encoder- ?physical laws?
Message source- ???
Message- mathematics

or something like that- just an example. This seems like a very interesting and fruitful route to take. It has already given me a funny idea- noisy shadows :rofl: Well, perhaps you had to be there.

[hypnagogue]

In what senses can mathematics be said to be a human invention? Clearly the formal systems we use to describe mathematical relationships is a purely human invention, a peculiar kind of language. However, for all of mathematics to be a purely human invention, I believe it would have to be the case that no mathematical relationships could be said to be pre-existing or fundamental aspects of reality itself, and this I take to be clearly false. In this sense, at least the basic phenomenon of mathematical relationship in general must have been discovered, not invented.

There is another, more abstract sense in which much of mathematics is discovery and not invention. Suppose a mathematician devises some odd-ball axioms within a specified formal system (call this conglomeration F) with no obvious relationship to physical reality. So far, it is uncontroversial that these specific axioms in this specific system are just abstract inventions. But now the mathematician does some fiddling around and derives some new theorems from F. These theorems are just a sort of restatement of information already contained within F and are thus, in a sense, equally as much inventions as F is. However, the mere act of 'unfolding' F, or making explicit information that was once implicit within it, is a sort of discovery in its own right. By way of analogy, I can haphazardly create (invent) a random sculpture out of clay, but after my initial act of creation I can still go about discovering new aspects of my sculpture by rotating it and looking at it from different angles.

Thus I think that there is a significant sense in which much of mathematics can be said to be discovery, not in the sense of discovering metaphysical Platonic forms, but rather properties of nature and properties of abstract systems. There is a dynamic interplay between the two, where discovery spurs invention and invention spurs discovery, where discovery is generally a discovery of completely abstract relationships that can sometimes be shown to exist in nature.

[Royce]

For once we agree, hypnagogue, but I think that you put it a little better than I. Your point that even in a pure human invention there may be unknown implication or features that are inherent in the system and later discoveredis a good one that I was trying to point out.
My, and Penrose's, main point is that it is nature and the study of nature that leads us to discover mathematics and the intricacies in the system.

[12345]

i wouls asume that it has been invented.

[Royce]

i wouls asume that it has been invented.

Why? How?
Why do you assume that it was invented?
How was it invented without discovering first the mathematical relationships in nature and then inventing a symbolic mathematical language to describe these natural relationships.
Mathematical relationships exist in nature and are observed by us. We invented the symbolic mathematical language so that we don't have to have oranges and/or apples on hand every time we want to perform a mathematical operation just like we invented algebra and the use of letters to represent numbers either as unknowns or as unspecified i.e. "x" ,A+B=C, 1,2,3...n etc.

The symbols and language of mathematics is beyond doubt an invention or convention but mathematics itself is a discovery of nature. Mathematical relationships exist in nature, the universe, with or without mankind. At least this is the position of Penrose and others including myself since reading his book.

[Dooga Blackrazor]

How do you invent something without first discovering how it is to be invented? I'd say it was discovered.

You discover how to invent something don't you? Therefore everything is discovered? Discovery and invention often come together in my book.

[selfAdjoint]

How do you invent something without first discovering how it is to be invented? I'd say it was discovered.

You discover how to invent something don't you? Therefore everything is discovered? Discovery and invention often come together in my book.

But if you discover something, it must have existed before you found it, no? Otherwise if by discovering it you made it newly exist, that would be invention wouldn't it? Did the Pythagoren Theorem or Wile's proof of Fermat's theorem exist before human beings conceived them? Where and how did they exist?

[hypnagogue]

Did the Pythagoren Theorem or Wile's proof of Fermat's theorem exist before human beings conceived them? Where and how did they exist?

Of course they did not exist as theorems, but surely their truth values existed. For instance, the Pythagoeran Theorem existed before its formal discovery insofar as anyone who happened to construct squares from the sides of a right triangle would find that the area of the square of the hypotenuse would equal the area of the other two squares combined. The name and formalism expressing the Pythagorean Theorem are human inventions, but the mathematical relationship it describes in nature is not-- that mathematical relationship existed before any human conceived of it.

[selfAdjoint]

For instance, the Pythagoeran Theorem existed before its formal discovery insofar as anyone who happened to construct squares from the sides of a right triangle would find that the area of the square of the hypotenuse would equal the area of the other two squares combined.

Precisely my point. The "truth" doesn't exist apart from human activity. The non-sensate universe has no "truth", just facts.

[hypnagogue]

Precisely my point. The "truth" doesn't exist apart from human activity. The non-sensate universe has no "truth", just facts.

Then humans discovered a fact in nature and named it the Pythagorean theorem. I don't see a big difference.

[Royce]

Then humans discovered a fact in nature and named it the Pythagorean theorem. I don't see a big difference.

This, of course, is Penrose's and now my point. The facts, mathematical relationships exist in nature, in the universe, in reality, however you may want to put it since the beginning and mankind discovers these relationships, facts. We invented the symobols and language or better adopted them by convention to discribe the "Natural" facts and relationships that we discover by observation and investigation of nature.
Given a circle, pi exists. Given a triangle, the Pythagorean theorem exists etc.

[Janitor]

If the first twenty spacefaring alien species that we earthlings come into communicative contact with all tell us that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the sides--making allowances for differences in language and symbology of course--then it would have to be a pretty obstinate person to say that all of mathematics is the free invention of the human mind, and nothing more than that. Or so it seems to me.

[12345]

:biggrin: Why? How?
Why do you assume that it was invented?
How was it invented without discovering first the mathematical relationships in nature and then inventing a symbolic mathematical language to describe these natural relationships.
Mathematical relationships exist in nature and are observed by us. We invented the symbolic mathematical language so that we don't have to have oranges and/or apples on hand every time we want to perform a mathematical operation just like we invented algebra and the use of letters to represent numbers either as unknowns or as unspecified i.e. "x" ,A+B=C, 1,2,3...n etc.

The symbols and language of mathematics is beyond doubt an invention or convention but mathematics itself is a discovery of nature. Mathematical relationships exist in nature, the universe, with or without mankind. At least this is the position of Penrose and others including myself since reading his book.

exactly......i should be more specific about what i mean. i meant the number system, and the other mathematical terms have been invented by man. i believe that every thing in nature is relative to an equation.

[matt grime]

But Pythagoras's theorem is only true in Euclidean Geometry. It certainly isn't true in other geometries (which are more natural, despite what people think), and is a consequnce of the *definition* of geodesic.

[honestrosewater]

It seems that by some of these arguments, humans have never invented anything; since any invention (with the usual stipulations that an invention be intelligible and useful) is the result of observations. Surely the lightbulb existed in some form in nature, in however many fractions, but it required a kind of thought that differs from "mere" discovery. Discovery and invention may or may not require intent, yet that matters little here. Discovery entails only finding or happening upon something and does not include making modifications to or extractions from the thing discovered. The latter is invention.
Well, that's MHO anyway.
Happy thoughts
Rachel

[sol2]

Some like to watch the steering of boids (http://www.red3d.com/cwr/boids/):), and some like to watch the Dance of the Honey Bee (http://www.physicsforums.com/showpost.php?p=223512&postcount=5):)

Heck, why not refer to John Nash again here, and the principals of negotiation? The consequence of conceptualization had its basis in mathematical interpretation. So given the gifted eye of the mathematicain, how watchful is s/he of the anomalies in nature?

[honestrosewater]

If you want to delve into a mathematician's brain, why not deny the invention of mathematical language? That is certainly more reasonable than denying the invention of mathematical concepts.

Language, along with it's cognitive structures, has a longer history and wider usage, even amongst nonhuman animals. The seemingly innate mathematical abilities lose their mathematical character when viewed as the accompaniments or by-products of already developed liguistic cognitive structures. How does simple counting differ from putting a name to a face? But there is more to math than putting two and two together. (And I have yet to see those monkeys on typewriters write Shakespeare BTW :wink: )

From denying mathematicians credit for their conscious intent and manipulation of ideas follows the elimination of the entire category of human invention, thus making the entire invention/discovery distinction pointless anyway. Perhaps the "real" question is then, "Conscious or mechanistic"?

Sorry, I'm tired and a bit grumpy.
Happy thoughts
Rachel

[hypnagogue]

It seems that by some of these arguments, humans have never invented anything; since any invention (with the usual stipulations that an invention be intelligible and useful) is the result of observations. Surely the lightbulb existed in some form in nature, in however many fractions, but it required a kind of thought that differs from "mere" discovery. Discovery and invention may or may not require intent, yet that matters little here. Discovery entails only finding or happening upon something and does not include making modifications to or extractions from the thing discovered. The latter is invention.
Well, that's MHO anyway.
Happy thoughts
Rachel

The way I take the phrasing of this problem, something that is invented is something that is completely novel (ie, had not yet existed until its invention), whereas something that is discovered is something that already existed.

In the case of math, it's obvious that the specific formal languages and so on that we use to describe mathematical relationships are just human invention. However, to say that math is a whole is an entirely human invention would mean that mathematical relationships do not really exist in nature. If this is the case, it's not clear that Newton would have ever been able to write his Principia.

[hypnagogue]

But Pythagoras's theorem is only true in Euclidean Geometry. It certainly isn't true in other geometries (which are more natural, despite what people think), and is a consequnce of the *definition* of geodesic.

If you draw two perpendicular lines measuring 3 and 4 inches respectively, then a third line connecting the two non-intersecting endpoints will always be 5 inches. Try it. :biggrin:

There is a question of domain of applicability-- from a QM perspective, on a small enough scale it would probably be impossible to construct a right triangle in the first place. But in our familiar human scale corner of nature, where continuity and Euclidean space are great approximations, the Pythagorean theorem will not fail to hold. Therefore the Pythagorean theorem describes a mathematical relationship that exists in our familiar niche of nature, even if it fails to hold in more exotic circumstances.

[honestrosewater]

The way I take the phrasing of this problem, something that is invented is something that is completely novel (ie, had not yet existed until its invention), whereas something that is discovered is something that already existed.

*Completely* novel? I think it is exactly this definition which excludes the possibility of invention. Existed in what form- in the same exact form as the invention? I think invention is higher up the totem pole. An invention being based in reality does not exclude it from being an invention.
Is there any thought in your head or any work of your hands that is *completely* your own and not based on anything else?

In the case of math, it's obvious that the specific formal languages and so on that we use to describe mathematical relationships are just human invention. However, to say that math is a whole is an entirely human invention would mean that mathematical relationships do not really exist in nature. If this is the case, it's not clear that Newton would have ever been able to write his Principia.

How so?! Language is not *completely* novel. Language evolved from cave paintings, pictographs, etc. which were certainly based on observations of nature. The language and the concepts can have similar bases. But the observation of a ladder leaning against a wall and the concept of the relative sides & angles of a perfect right triangle are two different things. They are as different as Chopsticks and Chopin, if not more so.

Happy thoughts
Rachel

[matt grime]

If you draw two perpendicular lines measuring 3 and 4 inches respectively, then a third line connecting the two non-intersecting endpoints will always be 5 inches. Try it.

No it isn't 5 inches exactly. Firstly real world measurements don't take that accuracy, and secondly you're ignoring the (possible) curvature of space-time. And by using the word measure you are eliding the issue of in which geometry your geodesics are defined.

[aqualine]

The argument has gone in several directions. i think the question is not about "math", but the substance that math asserts to delineate, and whether those motions are real-- and disregarding of human note-- and so "discovered", or completely dependant upon that human notice and non-existant without it: invented. But then again, "math" as a construct is really inextricable from the mechanics it denotes in as much as it denotes those mechanics and no others. So it is incomplete; but that does not make it necessarily unreal. You could say that if it is incomplete it is relatively finite and if finite, infinitely irrelevant. But again, an irrelevant mechanism is not necessarily a non-real mechanism. Crudely, were you to need to cross an ocean, you would have to take a path to get there. But is that path real, or would it die with the need to cross? Certainly mechanically there remains its possibility and physically there were even its effect. And there is the real question: Is the cause/ effect relationship real or does it hang on perception? Two particles in space could be seen to be on a path to collide, and we can induct that they will collide. But it is possible to imagine (in the same way that it is possible to imagine nothing or everything) a sort of omniscopic point of view whereat they are not approximating, but travelling apart. That would make the initial perspective, along with its perceived mechanism, not irrelevant, but have been all along unreal. That, of course, would appear to require more than a single "math"-- neither one ending with a dynamic or inequality, nor one ending with a static or equality, but both without regard for time-- and that is not possible as far as we are concerned. But that does not make it impossible. Our concern with its math does not make it impossible that our math with its mechanics is perfectly unreal.

[robert Ihnot]

No it isn't 5 inches exactly. Firstly real world measurements don't take that accuracy, and secondly you're ignoring the (possible) curvature of space-time. And by using the word measure you are eliding the issue of in which geometry your geodesics are defined.

The Egyptians had to constantlyl survey the land as the Nile frequently overflowed. Try telling them your bull****!

Math was not well-developed in those days, but the Egyptians had practical abilities. That is how they survived.

[robert Ihnot]

If you want to delve into a mathematician's brain, why not deny the invention of mathematical language? That is certainly more reasonable than denying the invention of mathematical concepts.

Language, along with it's cognitive structures, has a longer history and wider usage, even amongst nonhuman animals. The seemingly innate mathematical abilities lose their mathematical character when viewed as the accompaniments or by-products of already developed liguistic cognitive structures. How does simple counting differ from putting a name to a face? But there is more to math than putting two and two together. (And I have yet to see those monkeys on typewriters write Shakespeare BTW :wink: )

From denying mathematicians credit for their conscious intent and manipulation of ideas follows the elimination of the entire category of human invention, thus making the entire invention/discovery distinction pointless anyway. Perhaps the "real" question is then, "Conscious or mechanistic"?

Sorry, I'm tired and a bit grumpy.
Happy thoughts
Rachel

It does seem in a way that language is more innate. Clearly the practicality of language is enormous. Math is a little different. Sure, there is reason to know how to count, but that ability like writing was often left up to scribes who recorded cattle sales, etc. Detailed math begain as the job of the specialist.

Clearly, people can see that a picture represents a thing, not itself. Animals, maybe can not. Without this abstract ability nothing much would have been posible as civilization progressed.

But, with math, such as tensor analysis or 4 dimensional space, how on earth would such math ability help most people? Thus math ability, real abstract ability, which is rare, seems to be a special gift of the gods that to a primitative society would be meaningless.

Thus a new dimension of the question between invention and discovery is how are mathematicians able to preceive, to become cognative of sophisticated mathematical structure at all?

This seems to be really a more important question. After all, what society can not preceive or undersand can't hurt you, or anyway, no one wouldl know the difference!

To go over this question again: What use would a savage have of Relativity? What possible survival value would there be in such advanced thinking? Thus how could the ability to understand that develope at all?
roberteignot.

[p-brane]

I've been reading Roger Penrose's " The Emperor's New Mind." He brought up a question that I once thought I knew the answer. "Is Mathematics invented or discovered?" I.E. Is Mathematics a purely mental construct or does(do) Mathematics exist, as in Plato's forms, somewhere, as Truths of Reality that we discover rather than invent?

And, by extension is Logic invented or discovered truths of the universe?

Think about it. Then give us your thoughts on the matter. It is not really as simple or straight forward as it first appears.

Math certainly exists now.

I would say that math is a tool that was waiting to be discovered and used...... much like a stick that was discovered and used to coax ants out of an ant hill by a bird or a chimpanzee.

[TeV]

Math certainly exists now.

I would say that math is a tool that was waiting to be discovered and used......I certainly can't claim it exists since when you speaking about existance,it must be carefully clarified what it means for abstractive terms.But,wether we found that "tool" somehow implemented in world around us,or order in it had just inspired us to invent the "tool",something is interesting:There is tight connection between math development and physics development (the science describing world around us).In other words the language of physics uses language of math.Not just technically,that wouldn't be so weird.It is nearly parallel in History.Many abstract math theory "structures" become sooner or later part of some branch in physics.
Personally,the most amazing thing to me is we managed to mathematically formulate laws of quantuum theories that work so well ,while not knowing/having clear idea why Nature behaves at quantum level in so strange way that there are many interpretations of QM driving people crazy.

[sol2]

I certainly can't claim it exists since when you speaking about existance,it must be carefully clarified what it means for abstractive terms.But,wether we found that "tool" somehow implemented in world around us,or order in it had just inspired us to invent the "tool",something is interesting:There is tight connection between math development and physics development (the science describing world around us).In other words the language of physics uses language of math.Not just technically,that wouldn't be so weird.It is nearly parallel in History.Many abstract math theory "structures" become sooner or later part of some branch in physics.
Personally,the most amazing thing to me is we managed to mathematically formulate laws of quantuum theories that work so well ,while not knowing/having clear idea why Nature behaves at quantum level in so strange way that there are many interpretations of QM driving people crazy.

There is always a graduation in thinking from our predeccesors?

Reimann did it having consumed Gauss, and Saccheri.

There were save assumptions in a euclidean world, and now, the consistancy of the math is evolving. Imagine, Topological considerations in a such an abtract world. :smile:

See Kleins ordering of geometries (http://superstringtheory.com/forum/geomboard/messages4/18.html)

I think this is Einsteins lesson about the origins of interpretation? That it must be geometrical defined. If this is the case then we must have some origins from which to begin?

[TeV]

There is always a graduation in thinking from our predeccesors?
.....
Einsteins lesson about the origins of interpretation? That it must be geometrical defined. If this is the case then we must have some origins from which to begin?Yes.What would we be without our predeccesors?Newton himself said he had seen further becouse he had been on the shoulders of the "giants".
As I have heard of Einstein's standpoints on the philosophy of physics ,it was sort of speak "geometrical".He even liked ,believe it or not,Platon's idea where universe was just empty space and bodies in it are chunks of empty space separated by geometrical surfaces.

[sol2]

Yes.What would we be without our predeccesors?Newton himself said he had seen further becouse he had been on the shoulders of the "giants".
As I have heard of Einstein's standpoints on the philosophy of physics ,it was sort of speak "geometrical".He even liked ,believe it or not,Platon's idea where universe was just empty space and bodies in it are chunks of empty space separated by geometrical surfaces.

Plato's solids? Where does it end? Bucky Ball's or crystallography? Fractals?

So we see this shift on Pythagorean harmonies as a extension of string theory and Lqg on the other hand? There still is this struggle to define on the issues of background and non background.


What do we mean when we say "continuum"? Here's a description Albert Einstein gave on p. 83 of his Relativity: The Special and the General Theory:

The surface of a marble table is spread out in front of me. I can get from any one point on this table to any other point by passing continuously from one point to a "neighboring" one, and repeating this process a (large) number of times, or, in other words, by going from point to point without executing "jumps." I am sure the reader will appreciate with sufficient clearness what I mean here by "neighbouring" and by "jumps" (if he is not too pedantic). We express this property of the surface by describing the latter as a continuum.

The Continuum Hypothesis (http://superstringtheory.com/forum/geomboard/messages2/117.html)

So you can see how this issue can materialize respective positions?

This directed my attention to how all maths arise, and the source from which they materialize. If it is inherent in observation that nature will supply us this definition, then how shall we describe it? :smile:

Platonic Solids and Plato's Theory of Everything

The Socratic tradition was not particularly congenial to mathematics
(as may be gathered from A More Immortal Atlas), but it seems that
Plato gained an appreciation for mathematics after a series of
conversations with his friend Archytas in 388 BC. One of the things
that most caught Plato's imagination was the existence and uniqueness
of what are now called the five "Platonic solids". It's uncertain who
first described all five of these shapes - it may have been the early
Pythagoreans - but some sources (including Euclid) indicate that
Theaetetus (another friend of Plato's) wrote the first complete account
of the five regular solids. Presumably this formed the basis of the
constructions of the Platonic solids that constitute the concluding
Book XIII of Euclid's Elements.

In any case, Plato was mightily impressed by these five definite shapes
that constitute the only perfectly symmetrical arrangements of a set
of (non-planar) points in space, and late in life he expounded a
complete "theory of everything" (in the treatise called Timaeus) based
explicitly on these five solids. Interestingly, almost 2000 years
later, Johannes Kepler was similarly fascinated by these five shapes,
and developed his own cosmology from them.

To achieve perfect symmetry between the vertices, it's clear that
each face of a regular polyhedron must be a regular polygon, and all
the faces must be identical. So, Theaetetus first considered what
solids could be constructed with only equilateral triangle faces. If
only two triangles meet at a vertex, they must obviously be co-planar,
so to make a solid we must have at least three triangles meeting at
each vertex. Obviously when we have arranged three equilateral
triangles in this way, their bases form another equilateral triangle,
so we have a completely symmetrical solid figure with four faces,
called the tetrahedron, illustrated below.

http://www.mathpages.com/home/kmath096.htm

You could see how useful the monte carlo method might be in describing quantum gravity in such a model of triangulation?

http://superstringtheory.com/basics/gifs/SinPlot.gif

Or Pythagoras as the first string theorist?


For additional Information (http://wc0.worldcrossing.com/WebX?14@194.h1WobsZmbR1.25@.1dde7b2e) Do not forget to scroll down immediately.

[matt grime]

The Egyptians had to constantlyl survey the land as the Nile frequently overflowed. Try telling them your bull****!

Math was not well-developed in those days, but the Egyptians had practical abilities. That is how they survived.



you're confusing the practical with the theoretical. the idealized world in which pythagoras's theorem is true is eulcidean geometery, which is reasonably approximated by the earth's surface locally. key word approximated. or are you a flat earther as well who hasn't noticed the natural geometry of the surface is spherical? mind you you also seem to think matter is infintely divisible too, so who knows what crackpot theories you hold.

if another alien speices did not view the universe in the same manner as we (which was the original point in question), perhaps they would not have developed an axiomatic geometry which had euclidean geometry as a model.

[Royce]

you're confusing the practical with the theoretical. the idealized world in which Pythagoras's theorem is true is Euclidean geometry, which is reasonably approximated by the earth's surface locally. key word approximated. or are you a flat earther as well who hasn't noticed the natural geometry of the surface is spherical? mind you you also seem to think matter is infinitely divisible too, so who knows what crackpot theories you hold.

if another alien species did not view the universe in the same manner as we (which was the original point in question), perhaps they would not have developed an axiomatic geometry which had euclidean geometry as a model.

Another term for euclidean geometry is plane geometry the geometry of a flat plane as opposed to spherical geometry or any other curved space or plane. On a flat plane all of Euclid's axioms hold true and cannot be denied. Any species who can imagine or experiences a flat plane will by necessity develop euclidean or plane geometry with all of the relationships being the exact same, thus supporting the position that mathematics is discovered rather than invented.
The same holds true for any curved space or surface. The relationships will remain exactly the same as they, the relationships are intrinsic properties of the surface at hand. There are no other possibilities!
It is then, as I said, axiomatic that mathematics are discovered properties of nature, reality or the universe, whichever you prefer, and not pure abstract constructions of our minds.
Try to describe the motion of a falling body in a gravitational field using any mathematics other than the one that we use now and learned in Physics 101.
Calculus is a natural and logical result of any such attempt. Newton and others did not invent calculus purely out of thin air but were led to it virtually by the hand out of necessity to describe such motion and other such phenomena.
While it is true that relativity and curved space can and does effect the results of such calculations it is only at the extremes that they have any significant effect at all. In a practical sense here on earth they can be safely ignored except when traveling great distances of the surface of the earth.
Then spherical geometry is the applicable math to use. The old adage of using the right tool for the right job applies here as everywhere. I would not use a sledge hammer to attach a glass plane to a window frame any more than I would use pane geometry to describe the surface of a sphere

[sol2]

I liked your description Royce.

There are reasons why Enstein choose a positive curvature in regards to Reinmann's sphere?

I can't help but be enamoured with the relationship of people like Sachheri, Gauss, and Reinmann, but this by know means demonstrates the geneology of this developement, but reminds us of the issues with Bolyai, Minkowski and Lorentz.

Moving to hyperdimensional realities is as much the acceptance of the world of Gauss, as it is to see how Einstein evolved this move and was futher extended in the visions of Klein. Imagine a cylinder, but before this Mercuries orbits, Taylor and Hulse and then the Bose Nova? :smile:

Indeed the need for this consistancy is extremely important as I believe it has to form the basis of understanding the move to quantum geometry. Einstein just didn't understand the geometry but raised the issue of gravity in GR. We are being lead along here I think when we consider Kaluza and Klein, and the unification of electromagnetism with Gravity. There seems to be no other avenue as far as I can tell where such consistancy demonstrates it's uniquesness, not just with the geometrical evolution, but also demonstrates a call for a experimental justification.

That's just my point of view though :smile:

[TeV]

On a flat plane all of Euclid's axioms hold true and cannot be denied. Any species who can imagine or experiences a flat plane will by necessity develop euclidean or plane geometry with all of the relationships being the exact same...
The same holds true for any curved space or surface. The relationships will remain exactly the same as they, the relationships are intrinsic properties of the surface at hand. There are no other possibilities!
Well,Euclid's axioms hold true on flat plane becouse they are defined as axioms.Guided by everyday experience people (in world around them) picked some "obvious" truths and formed minimum set of axioms that satisfied logical requirement that no paradox can be derived from them.Axioms ,of course,cannot be proved or disproved.
On the other hand,Lobacevski showed that one of these axioms (axiom of paralels) can be altered on the very same flat plane to produce new logically valid geometry.Representations of these geometries in 3D Cartesian system generates "curved" sufaces (pseudospheres etc..) ,though.

[BobG]

Yes.What would we be without our predeccesors?Newton himself said he had seen further becouse he had been on the shoulders of the "giants".

That's a quote with a funny history. Newton said that as a way of closing one of his letters to Robert Hooke. Newton totally despised Hooke, so much so that it controlled quite a few of his actions and caused him a lot of problems. So much so, that in spite of Hooke's legendary obnoxiousness, you had to have some doubts about Newton's emotionaly stability, as well. Hooke also suffered from some physical handicaps, so much so that people compared him to a dwarf. Even if a veiled insult, it has some irony. Newton's Principia was at least partially motivated by a disagreement Newton had with Hooke.

Since I should say something to keep this on topic, how about a quote from Leo Kronecker, "God created the integers, all else is the work of man."

Contrary to what a previous post implied, only humans can 'count'. All animals, including humans, have some innate ability to descriminate between a certain amount of objects without having to resort to counting. Humans are supposed to be able to recognize up to 4 objects without counting (which would explain why, when tallying things with marks, we tend to use a slash through the previous 4 marks to indicate 5 and why 5 is such a recurring base in the world's numbering systems). Counting, the first mathematical development, is an invention of man that enables him to extend his natural abilities beyond just four. (I guess Leo wasn't quite right - maybe God just created the first four integers)

And while math was invented to describe relationships between different things in the physical world, the math isn't the things described. For example, what is the ratio between the circumference of a circle and the diameter of a circle? The relationship exists for every circle and the math we invented can approximately describe that ratio, but math isn't going to give you the 'real' ratio. There are many relationships in nature that can't be expressed, only approximated, by numbers or mathematical equations.

As such, math is a logical language invented by mankind. Like all other human languages, its is easily expandable and very adaptable. In the same manner one might 'discover' the similarities between the Sun and a red rubber ball and invent a new 'simile', mathematicians and scientists can 'discover' how adept previosly invented mathematical concepts are at describing newly discovered phenomenon.

[TeV]

************************************************** **
Contrary to what a previous post implied, only humans can 'count'. All animals, including humans, have some innate ability to descriminate between a certain amount of objects without having to resort to counting. Humans are supposed to be able to recognize up to 4 objects without counting..
************************************************** **
Hmm,I thought It was scientificly proven that some animals can "count" or better say recognize set of object up to about 4 too.
Crows and dolphins in particular .But don't take me for word,I'm not sure.

[sol2]

Maybe the Bee's can count considering their dance (http://www.physicsforums.com/showpost.php?p=223512&postcount=5)? :smile:

Oh I forgot to mention Lobachevsky (http://superstringtheory.com/forum/geomboard/messages/27.html) and since TEV did, I too acknowledge his contribution.

[Gza]

Here's what I think mathematics is. I'm not sure of the philosophical sophistication of the people posting here, but there is a concept know in philosophy as "background" that is important to understand my idea. It is basically the unavoidable basis that you are required to base any opinion you have about the world upon.(an example of background would be in our visualization of more than three dimensions. The operational nature of our three dimensional processing visual cortex serves as the background to our vision of geometry. We cannot make a sensory based visualization of geometry in more than 3 dimensions, so our visual cortex limits us, and thus serves as the background in this analogy. We must base all visual opinions against this background, and can never overcome it as a consequence of our biological structure.)

I could also analogize it mathematically, bringing it closer within context, as a the basis vectors that span a vector space. You can make any judgement of the world, but all these judgements are nothing more than some simple or complex "linear combinations" of background "basis vectors". (I really hate to define mathematics in terms of itself, but bear with me.) Math to me is the actual study of background, abstracted away from the various ways it can be scaled and combined within its own background space. We are determining the rules for construction of the background basis, so that we are free to "build up" all the logical combinations possible. You could also say my entire definition is equivalent to saying nothing more than mathematics is an axiomatic, deductive brance of reasoning, but I think this definition is stronger than that. The deepest axioms we can come up with are the ones derived from background limitations.

[p-brane]

I certainly can't claim it exists since when you speaking about existance,it must be carefully clarified what it means for abstractive terms.But,wether we found that "tool" somehow implemented in world around us,or order in it had just inspired us to invent the "tool",something is interesting:There is tight connection between math development and physics development (the science describing world around us).In other words the language of physics uses language of math.Not just technically,that wouldn't be so weird.It is nearly parallel in History.Many abstract math theory "structures" become sooner or later part of some branch in physics.
Personally,the most amazing thing to me is we managed to mathematically formulate laws of quantuum theories that work so well ,while not knowing/having clear idea why Nature behaves at quantum level in so strange way that there are many interpretations of QM driving people crazy.

Yes, you have several points there. I believe math to be a language developed to interpret our physical surroundings and our observations of the same. That would explain the tandem effect seen in the parallel evolution of math and physics, which you have pointed out.

Sometimes the language of math gets ahead of the observations we make so that we are witnessing a description of something that has not been actually observed. Not unlike predictive science fiction. Then that bit of language becomes validated by a physical occurance. Take Einstein's prediction of black holes for example.

But, in the end I would say that math is and always will be a simple description we have devised and/or discovered. We use math as an overlay on the functions we find in our natural surroundings. We take our observations and we conveniently package them in mathmatical formuli and they remain as records and testament to what we have seen in nature or to what we can calculate as being possible in nature.

The proof that math is a relative interpretation of nature is in the fact that it is a tool that can be used in a large variety of situations. Its usefulness always remains dependent, relatively, on the requirements and nature of those situations.

[selfAdjoint]

But what about all the abstract math that never gets applied to physics? You can assert that it all will be ultimately relevant somehow (I have heard mathematicians do this), but that is faith in things unseen. I think mathematics is the free excercise of some mental abilities we are born with. The more I think about Chomsky's new theory that recursion pure and simple underlies our language skills, the more I suspect that it underlies mathematics as well.

[matt grime]

Another term for euclidean geometry is plane geometry the geometry of a flat plane as opposed to spherical geometry or any other curved space or plane. On a flat plane all of Euclid's axioms hold true and cannot be denied. Any species who can imagine or experiences a flat plane will by necessity develop euclidean or plane geometry with all of the relationships being the exact same, thus supporting the position that mathematics is discovered rather than invented.
The same holds true for any curved space or surface. The relationships will remain exactly the same as they, the relationships are intrinsic properties of the surface at hand. There are no other possibilities!
It is then, as I said, axiomatic that mathematics are discovered properties of nature, reality or the universe, whichever you prefer, and not pure abstract constructions of our minds.
Try to describe the motion of a falling body in a gravitational field using any mathematics other than the one that we use now and learned in Physics 101.
Calculus is a natural and logical result of any such attempt. Newton and others did not invent calculus purely out of thin air but were led to it virtually by the hand out of necessity to describe such motion and other such phenomena.
While it is true that relativity and curved space can and does effect the results of such calculations it is only at the extremes that they have any significant effect at all. In a practical sense here on earth they can be safely ignored except when traveling great distances of the surface of the earth.
Then spherical geometry is the applicable math to use. The old adage of using the right tool for the right job applies here as everywhere. I would not use a sledge hammer to attach a glass plane to a window frame any more than I would use pane geometry to describe the surface of a sphere


As you admit in this post plane geometry is only an attempt to model that which we see locally. That does not support the idea that it was discovered, and in fact surely it demonstrates that we invented it? The fact that we can take the parallel postulate as true or false without contradicting the other axioms also backs up this assertion.

[BobG]

Calculus is a natural and logical result of any such attempt. Newton and others did not invent calculus purely out of thin air but were led to it virtually by the hand out of necessity to describe such motion and other such phenomena.

Maybe it's almost just semantics. To be 'mathematics', it has to have some set laws and constraints to keep it internally consistent - as opposed to the real physical world, which sometimes yields some unexpected exceptions. Not all math is based on real observations, but sometimes just contrived so that 1+1 doesn't just usually equal 2, or usually equal some value around 2 - it always equals exactly 2, no exceptions. (In the exponential function, a^x, why does 'a' always have to be greater than zero and not equal to 1 - what real world phenomena does that relate to? You mean I can't make any boxes that are 1 meter in length, 1 meter in width, and 1 meter in heighth? Then I guess it's a good thing all my boxes are 39.37 inches by 39.37 inches by 39.37 inches!).

Newton discovered a relationship about the slope of the tangent to a curve plotted on a graph. But, even the math he invented based on this relationship wasn't exactly accepted. In fact, quite a few ridiculed it - you can't have instantaneous velocity, because, if time is zero, then there is no motion (poor Zeno was rolling around in his grave). Plus, you can't divide by zero - it's a mathematical law! It still needed quite a bit more work (add some more laws, constraints, etc - specifically, somebody had to invent 'limits') to seal up the 'holes' in it before it 'officially' met the standards of being a branch of 'mathematics'.

The final 'invented' product was not the same as the 'discovered' relationship that inspired it. No real difference from most other inventions, which are almost always inspired by something observed in the physical world.

[TeV]

But what about all the abstract math that never gets applied to physics? You can assert that it all will be ultimately relevant somehow (I have heard mathematicians do this), but that is faith in things unseen. .. IMHO,there are SOME differences between abstract math that never gets applied to physics and abstract maths that gets.
My impression:simple reason might be that nature uses the most simple wayouts,paths and mathrules in the base of its' modus operandi,while mathematician sometimes gets astray creating his own rules.Some of these rules might not be too efficient in the basis for the big machine of the universe and Nature rejects them.Example:transfite numbers are rejected by Nature in advance.
That's the freedom of math.Physicist seeks and explores laws of Nature while mathematician creates his *own* rules of game.Note the difference between used words :laws and rules.
Interestingly and quite obvious :mathematicians are also part of nature,beings made of flesh ,water and bones,bulks of matter that are organized in functional system ,and evolutionary quite efficient from the standpoint of mother Nature.Therefore...

[matt grime]

"transfinite numbers are rejected by nature in advance"
what does that mean? what does it mean to be accepted, and why are transfinite numbers not accepted. admittedly there is not an infinite number of any one object in the universe, but even so, what's that got to to with it?

[p-brane]

As you admit in this post plane geometry is only an attempt to model that which we see locally. That does not support the idea that it was discovered, and in fact surely it demonstrates that we invented it? The fact that we can take the parallel postulate as true or false without contradicting the other axioms also backs up this assertion.

Here's where definitions of "discovery" and "invention" may come in handy.

When X-rays were discovered they were discovered, not invented. The various uses of X-rays was invented but not the X-ray itself.

When it was discovered that 1+1 objects equaled a group of 2 objects this discovery became one of the basises for several inventions in the realm of mathematical equasions.

The difference between discovering X-rays and discovering a pattern in grouped objects is that the objects and their "mathmatical" relationships are judged to be mathmatical by our interpretation, from our perspective. The X-rays are "X-rays" regardless of what we think of that particular electromagnetic spectrum.

In a way it is begining to look as though math is purely an invention, manifest of the imagination of over-cerebral humans, much like any other language. Thank you.

[TeV]

"transfinite numbers are rejected by nature in advance"
what does that mean? what does it mean to be accepted, and why are transfinite numbers not accepted. admittedly there is not an infinite number of any one object in the universeThere isn't infinite number of any object in the universe I agree,but there's potential infinity in EM and gravitational force reach for instance (according to currently accepted models of these force-field interactions).Also,this is just a possibility and interpretations depend on model of the universe and destiny of the same.Potential infinity isn't the same thing as actual.Hence,no actual infinity-no transfite numbers , ordinals etc.
In sense I hold the universe is sort of "constructivistic machine".
Of course,this is my opinion.You may agree or not.

[matt grime]

but TeV, why must numbers only be things which count physical objects? why can they not be used to enumerate the state(s) of a system? and as such there are an infinite number of possible states of a system.

[p-brane]

My guess is that there are as many numbers, states, systems and purposes as we can construe out of the material we have available to our five or more senses. This variety and magnatude of states includes our actual senses as well. Its a never ending fractal....... well, its never-ending until you stop looking into it. Then it ends.

Imagine that math goes away for the summer. There is sand but no one's counting the grains. There's an infinite party. Try to carry that thought into next fall. Cheers!

[TeV]

why can they not be used to enumerate the state(s) of a system? and as such there are an infinite number of possible states of a system.With finite number of physical objects,recognizing finite number of fundamental force interactions in system,there "is" only possibility* for one infinity-trough the time unlimiting operation procedure:the potential infinity.This is also supported by spacetime quantization requirement in modern era of the science.One can make yourself easier by introducing continuum description of space in aproximation ,but this is just a cheating (a very good one since growth and number of states of the evolving system is astronomical).For * see 3 posts up:we are not sure about the destiny of the universe.Therefore,since the ideas of continiuum and actual infinities are completely human that leaves the debate of the meaning of actual realization in nature fruitless.
But even,from the standpoint of potential infinity sign of singularities in any physical theory are signs of the *sickness* in theory.That was what I wanted to emphasize.