Zero zero ?

[disregardthat]

Maths is an exploration of the space of all possible forms using various representations of localised substance to construct every kind of shape that can be imagined.

Mathematics is not the exploration of anything. At least not literally (it is misleading to say so). It is something we create, and not more than what our mathematical calculus has been expanded into through logical inference. Mathematics is more like a symbolical machinery, a collection of algorithms.

We use mathematics to explore things we can imagine, and more tangible things. By postulating certain properties of concepts we can draw conclusions based on our mathematical calculus. It may very well be so that this is the 'purpose' of mathematics - if you want to put it that way (and I'll agree with you) - but it is not mathematics.

 

[apeiron]

Mathematics is not the exploration of anything. At least not literally (it is misleading to say so).

Perhaps you just misunderstand me. I was saying you make the Lego and then combine it every way it can be combined. You are exploring the phase space of the atomistic actions you have created. The terrain is unknown to you, but in some Platonic sense, it already exists. Much like very possible game of chess exists once the rules have been defined.

 

[Pythagorean]

The word 'mathematics' can mean different things to different people. Mathematicians are generally referring to the axioms, laymen are generally thinking about the numbers and symbols, scientists are generally referring to the discipline of mathematics as a study.

Personally, I think the axioms are invented. New axioms are discovered, but they are consequences of the original invention.

The symbols are obviously invented, but numbers like pi and e are most definitely discovered.

The discipline itself is obviously invented, but there is both discovery and invention taking place in the field.

 

[SonyAD]

The same way clay isn't fundamentally related to any object you decide to model with the clay.

Mathematics is logical clay.

I disagree. Mathematics had its origins in the need to cost effectively solve everyday practical problems. Like knowing how many apples you're bartering for a cow, how to divide a given number of loafs of bread equitably (as near to as possible) between a number of people, computing firing solutions, computing man-hours & labour force requirements, map making, figuring out how many different ways you can arrange stuff, etc.

That is what mathematics is, at its roots. It has its origins in practical necessities. Not pipe dreams about imaginary numbers and such.

The purpose is to be able to make models of reality with accurate logical statements; much like the purpose of making clay is for a sculptor to model. Some "claymakers" (mathematicians) DO just investigate formal rules to play with them, even though they don't have a meaningful physical counterpart.], but a lot of mathematics is driven directly by observation of the physical world.

I would say the purpose is to be able to make verifiable, reasonably accurate predictions about reality from reasonably accurate measurements. I don't think anybody really cares how stuff works as long as it does. I think this thread is evidence enough of that.

Do you see people questioning how irrational numbers can denote physical quantities? Nope.

Pythagorean put it well. Clay can be used to make sculptures of real things, but the clay itself is in no correspondence with what it imitates. The relation is always inferred from the outside. Furthermore, we don't even have to imitate real things at all.

I don't think the clay analogy is very good. At all. When I want to compute how many apples each of 5 people gets from a trolley cart full of them, I already know each one is bound to get less or all the apples in the cart? How do I know that? Math didn't tell me. It can't tell me.

How do I know no one can get more apples then there were in the cart initially? How do I know I have to divide and not multiply by the number of people? Or add the number of people to the number of apples? Or subtract from?

Nope. Sorry. Maths is just a dumb tool for use in making predictions about reality. It just models reality and does what you tell it to (by analysing the practical problem and deciding what operations to use, how to pipe them). When you tell it to do garbled nonsense the result is pointless.

I know to use division because I know it is the mathematical operation modelled after the action I perform in distributing the apples equitably.

Similarly, I know that by dividing the number of people by the number of apples in the cart I get the number/amount of people each apple gets, after an equitable distribution.

So how is mathematics not firmly rooted in reality? How was it not developed after and for reality (making predictions about it)?

There is nothing beyond that but insanity, as Georg Cantor may have found out if he realised he was going insane.

It's no secret that we use mathematics for various purposes like physical modeling and that it is developed for these things, but the important point is, which I have stated several times, that mathematics itself does not correspond to these things.

I don't see how you can end on that point. Again, how does addition not correspond to hoarding stuff in reality, for instance?

Mathematics is the purely formal development and use of strictly formal rules. It cannot correspond to anything.

It is rooted in observations about reality. It corresponds to reality. It went off the rails at some point, when the theoretical eggheads stole it from the engineers of their day.

However, that mathematics does not correspond to the real world does not imply that we have no motivation for the further development of mathematics, which you seem to suggest.

That is not what I suggest. What I suggest is that mathematicians try to develop practical maths with immediate, fundamental applications once in a while.

And that they try to stop needlessly delving in silliness, like using the complex plane instead of 2D vectors and whatnot.

How our calculations relates to reality is through an interpretation outside of mathematics.

No. That interpretation took place in the beginning and is what gave ous our particular flavour of mathematics, as you might put it, by defining its axioms. Where a + b does not equal 1 regardless of what a and b are, for instance. That interpretation is defining for and integral to mathematics.

It also takes place in the beginning of every new piece of mathematics developed. Like equations for computing the texture coordinates of the sample point from the texture coordinates of the triangles' tips by weighing these coordinates according to the distance to the sample point.

How could I have known to develop the math necessary for texture mapping, vertex rotations, fish eye lens projection, etc. on my own from scratch if what you say were true? How is it that they're basically the same others came up with long before myself (except I don't use matriceal representation), whose work I didn't have access to at the time?

So no, mathematics is not necessarily merely formal games without potential applications to reality, and this is because we have motivation for extra-mathematical use. That fact does not change the status of mathematics. At all.

What you're saying is basically that people developed imaginary numbers and group theory before the addition and subtraction of natural numbers for bartering. Abelian groups were just floating around in ethereal existence waiting to be plucked by some mathematician with spare time on their hands before anyone had even learnt to count.

One can, and one do occasionally, but one does not have to... Often we have a constructive application in mind for our use and development of mathematics. And often we don't, applications will often come as a 'side-effect' of the development of new mathematics, and there are many examples of this.

Yeah. Side effects like using complex numbers and the complex plane instead of 2D vectors. Or a Riemann sphere instead of polar projection.

I never said numbers were symbols, I said numerals were symbols, and they are. And I also said arithmetic is the formal manipulation of these symbols, and I can not see a single argument against that in your comment.

This is semantics. I don't know what you mean by numerals but numbers aren't symbols.

That mathematics deals only in formality means that the mathematical calculus is used and developed by following strictly formal well-defined rules. It's what I have been saying all along.

What strict, formal, well-defined rules did I follow when I developed my sign() function or fish-eye projection on my own?

As you can see in the link, we can formalize the use of what we call infinity as a symbol tied to certain rules; much like a number. And what you directly adressed; division of zero can also be formalized as shown. It puzzles me if you cannot see the connection between this and what I said right above the link.

To accomplish what? What do you accomplish by your formalisation of 1/0, infinity? Results based on division by 0, infinity. By hiding under an alias you just postponed the inevitable reckoning until you've done all the calculations you could. In the end, what you're left with is still very much as meaningless as it is still bound to division by 0 or infinity.

Or you can just make up some arbitrary convention like 1/0 = 2 and go from there. Still an exercise in pointlessness every bit as meaningless for making predictions about reality. Which has been the whole point of math since its inception.

We do have many different algebras as well. Some "useless", in that it has no current obvious application. In 'abstract algebra', addition is defined in many ways for different algebraic systems.

Why must we have a myriad of dud algebras instead of a myriad of

sillyAddition69(a,b) = 1
sillyAddition70(a,b) = a+b/2
sillyAddition71(a,b) = (a-1)×b
etc.

There is not 'one' algebra in the same way as there is not 'one' geometry.

Of course there is. And you can model and/or contain egghead brain farts inside the one geometry and the one algebra. :)

See above.

Why must I have a whole new (elliptical, hyperbolic) geometry to study curved surfaces (distances on them, angles, etc.)? Can't I model or study curved surfaces in "Euclidian" geometry?

Why do I need the complex plane? Don't I have vectors?

This is exactly what I'm talking about.

They are all studies of formalized structures. But they can also all have potential application outside of mathematics. That doesn't make them correspondent to whatever they might be used to represent, and it doesn't change the way we use mathematics. The use is always formal, completely rule-governed and without correspondence to physical reality.

I disagree.

 

[disregardthat]

Perhaps you just misunderstand me. I was saying you make the Lego and then combine it every way it can be combined. You are exploring the phase space of the atomistic actions you have created. The terrain is unknown to you, but in some Platonic sense, it already exists. Much like very possible game of chess exists once the rules have been defined.

Yes, I agree; exactly the way chess existed before it was invented. But it's an odd thing to say, isn't it? It should be just as odd to say it about mathematics. But for some reason it isn't. It's quite usual to state that we are discovering and exploring already existing mathematical structures, but that's as weird as saying that a carpenter is exploring the ways of ordering wood in space.

However, I will agree that in certain contexts the word discovery is more suitable than invention, but it must be clear that it really is invention/construction.

 

[disregardthat]

I don't think the clay analogy is very good. At all. When I want to compute how many apples each of 5 people gets from a trolley cart full of them, I already know each one is bound to get less or all the apples in the cart? How do I know that? Math didn't tell me. It can't tell me.

How do I know no one can get more apples then there were in the cart initially? How do I know I have to divide and not multiply by the number of people? Or add the number of people to the number of apples? Or subtract from?

Nope. Sorry. Maths is just a dumb tool for use in making predictions about reality. It just models reality and does what you tell it to (by analysing the practical problem and deciding what operations to use, how to pipe them). When you tell it to do garbled nonsense the result is pointless.

I know to use division because I know it is the mathematical operation modelled after the action I perform in distributing the apples equitably.

Similarly, I know that by dividing the number of people by the number of apples in the cart I get the number/amount of people each apple gets, after an equitable distribution.

So how is mathematics not firmly rooted in reality? How was it not developed after and for reality (making predictions about it)?

There is nothing beyond that but insanity, as Georg Cantor may have found out if he realised he was going insane.

This is nonsense. I can't make head or tail of your rambling.

It is rooted in observations about reality. It corresponds to reality. It went off the rails at some point, when the theoretical eggheads stole it from the engineers of their day.

The engineers were always aware of that their deductions were the strict use of formal rules when they resorted to their mathematical calculus. Much like anyone are when they e.g. try to solve a linear equation.

That is not what I suggest. What I suggest is that mathematicians try to develop practical maths with immediate, fundamental applications once in a while.

And that they try to stop needlessly delving in silliness, like using the complex plane instead of 2D vectors and whatnot.

This is silly. You are criticizing mathematicians for not developing useful mathematics.

What you're saying is basically that people developed imaginary numbers and group theory before the addition and subtraction of natural numbers for bartering. Abelian groups were just floating around in ethereal existence waiting to be plucked by some mathematician with spare time on their hands before anyone had even learnt to count.

No, I did not say that people developed imaginary numbers and group theory before addition and subtraction of natural numbers. That is a crazy assertion.

This is semantics. I don't know what you mean by numerals but numbers aren't symbols.

Of course it's semantics. We are having this discussion because you didn't understand my semantics. Check up the definition of 'numeral'.

What strict, formal, well-defined rules did I follow when I developed my sign() function or fish-eye projection on my own?

You create the sign() function by creating rules for inference. You use the sign() function when you use your already defined rules for inference. It's quite as simple as that.

To accomplish what? What do you accomplish by your formalisation of 1/0, infinity? Results based on division by 0, infinity. By hiding under an alias you just postponed the inevitable reckoning until you've done all the calculations you could. In the end, what you're left with is still very much as meaningless as it is still bound to division by 0 or infinity.

It's as 'meaningless' as any piece of mathematics, e.g. arithmetic. You will find it's 'meaning' in its extra-mathematical use. The projective plane is obviously useful outside of mathematics, so it's not meaningless.

Why must I have a whole new (elliptical, hyperbolic) geometry to study curved surfaces (distances on them, angles, etc.)? Can't I model or study curved surfaces in "Euclidian" geometry?

Can you model the spacetime of general relativity with euclidean geometry?



I can hardly see any arguments but random, non-sensical remarks in your post.

 

[Hurkyl]

I disagree. Mathematics had its origins in the need to cost effectively solve everyday practical problems. That is what mathematics is, at its roots. It has its origins in practical necessities.

Yes, yes, we all know that once upon a time, mathematics, science, and other branches of philosophy were all one great unified subject.

These days, we do a much better job of separating distinct subjects than the ancient Greeks did.

I would say the purpose is to be able to make verifiable, reasonably accurate predictions about reality from reasonably accurate measurements.

That sounds like physics.

Do you see people questioning how irrational numbers can denote physical quantities? Nope.

If you've never seen someone rant that there's no such thing as an irrational physical quantity, then you obviously haven't been on the internet long enough.

 

[Pythagorean]

I disagree. Mathematics had its origins in the need to cost effectively solve everyday practical problems. Like knowing how many apples you're bartering for a cow, how to divide a given number of loafs of bread equitably (as near to as possible) between a number of people, computing firing solutions, computing man-hours & labour force requirements, map making, figuring out how many different ways you can arrange stuff, etc.

That is what mathematics is, at its roots. It has its origins in practical necessities. Not pipe dreams about imaginary numbers and such.

So, nobody's arguing that? This doesn't confront the analogy whatsoever. In fact, it can be used as support for it. You're laying down the motivation for making logical clay in the first place.

I would say the purpose is to be able to make verifiable, reasonably accurate predictions about reality from reasonably accurate measurements. I don't think anybody really cares how stuff works as long as it does. I think this thread is evidence enough of that.

That's exactly what the clay analogy says. When we make a clay figurine of a woman, we don't expect it to bleed or bare children. It's a model that tells us about surface, volume, and shape: external spatial considerations about the woman its modeled after. And even then, it's not a 100% match, but it's more accurate than using hay, for instance.

Do you see people questioning how irrational numbers can denote physical quantities? Nope.

I'm not sure how this is at all relevant, but I'll respond as an aside:

Not as much nowadays. The Pythagoreans actually believed that irrational numbers didn't exist, and, if I recall correctly, tried to cover up their existence when they found out they were wrong.

 

[apeiron]

However, I will agree that in certain contexts the word discovery is more suitable than invention, but it must be clear that it really is invention/construction.

Not really because my purpose was to draw attention to what is actually invented - the axioms. Once you have invented these (based on intuitions and so derived from experience), then you can enter the platonic world you have just created and explore all its possible combinations.

The creative step is the first one step of chosing the axioms. Then the rest is inevitable as a formal consequence.

Of course, in real life mathematicians have to use a lot of imagination to discover these consequences. And even more to relate them back to the real world as it is experienced.

But the system of maths itself constrains such creative choice as much as possible - cordoning off contact with the messy real world as all the business that went up to chosing some axioms - then the rest can be just formal shut up and calculate. The divorce with reality which you think is so important (and it is) is justified on this basis.

So maths exists in its own invented Platonia because of this intellectual process. And to the degree that the axioms are truths about reality (which we cannot in principle know, but we can do an effective job on guessing via generalisation and abstraction) then reality will also be "platonic" - without actually being Platonic. Pi and e will also "exist out there" in a pragmatic sense. But again, maths is not completely Platonia because its foundations are our best guess axioms.

 

[disregardthat]

Not really because my purpose was to draw attention to what is actually invented - the axioms. Once you have invented these (based on intuitions and so derived from experience), then you can enter the platonic world you have just created and explore all its possible combinations.

I don't agree with the platonic perspective of mathematics. It is misleading in the way that it suggests that the mathematical structure is 'out there' waiting to be discovered. But mathematical structure is never 'out there', it comes into existence the moment we expand our mathematical calculus. At best it's a pedagogical picture of mathematics, as a sort of potentiality of mathematical expansion. But it gives us nothing.

As I mentioned; if the carpenter bought a given amount of material, is he exploring the way of ordering wood in space? No, he is constructing and inventing. The finished product was never 'out there'. The same thing applies for mathematical construction. pi and e were not more 'out there' than your chair was before it was constructed.

Sure, any proof is a necessary consequence of the axioms, but one is constructing these consequences by means of logical inference, much like a carpenter is constructing wooden objects (theorems) by means of his tools (logic) given his materials (axioms).

Axioms are not really special; indeed, in any statement of the form (A --> B), A serves as an axiom. But A doesn't have to have anything to do with reality or intuition. The axioms themselves are not inventions/constructions because they are not an expansion of the mathematical calculus. They serve as stepping stones for construction and invention. Like possible scenarios in a tactical plan.

Don't think of the set-theoretic axioms as 'the axioms' of mathematics. Set theory is just a model for mathematics (though set-theory is studied as a field itself), and more 'fundamental' axioms also come into play on other levels which does not mention set theory. Examples are the axioms of analysis; Axiom of Archimedes and the Fundamental Axiom of Analysis. Axioms are created all along the mathematical process. There is never a point in time at which we say "now we shut up and calculate". Even the 'foundational' axioms are constantly discussed, and there are alternatives.

 

[alt]

Hi Apeiron;

I thought I'd post this here, as we've mentioned Anaximander, and perhaps this relates to the thread 'Zero zero' in any case.

I'm reading a book where Anaximander is mentioned a lot. To quote;

First he proposed .. the Apeiron .., second he argued the earth floated freely in the sky .. third, life began in the sea, and fourth that the world as a whole came into existence and disappears periodically.

The fourth, as above, has go me stumped. I'm trying to understand what he acually meant by this, but can't find much on it. Are you able to elucidate ?

Thanks in advance.

 

[apeiron]

fourth that the world as a whole came into existence and disappears periodically.

Here Anaximander seems to be thinking of order that arises then falls apart again much in the fashion of a vortex in a stream or a dust devil. So the whole world arises out of the apeiron and self-organises to become a structure, but then eventually it can all fall apart and return whence it came, dissolving back into the apeiron.

It was a definite part of his philosophy that the apeiron was limitless and so would produce many worlds apart from our own (a multiverse view - except each universe is really just a solar system in scale). So he was simply predicting that our world could arise, and would eventually dissipate too.

 

[alt]

Here Anaximander seems to be thinking of order that arises then falls apart again much in the fashion of a vortex in a stream or a dust devil. So the whole world arises out of the apeiron and self-organises to become a structure, but then eventually it can all fall apart and return whence it came, dissolving back into the apeiron.

It was a definite part of his philosophy that the apeiron was limitless and so would produce many worlds apart from our own (a multiverse view - except each universe is really just a solar system in scale). So he was simply predicting that our world could arise, and would eventually dissipate too.

But periodically ? You will note the quote I included said 'disappears periodically'. Is this accurately reflecting what Anaximander was saying ? If so, he must have had a very long term view on things. Something like big bang / big crunch perhaps ?

Anyway, I haven't been able to find a page of his work where he says this. Would you have a link or post a transcript maybe ? In translation would be fine - in the original Greek would be finer too, as I can work through it.

Thanks for your help. Your explanation, above, was a start to my understanding of what he was positing. He did have some very novel thoughts.

 

[apeiron]

But periodically ? You will note the quote I included said 'disappears periodically'. Is this accurately reflecting what Anaximander was saying ?

I've never seen it claimed that he meant a cyclic scenario for "our" world. Only that he made two claims - the apeiron was infinite and so would be able to produce an unlimited number of worlds in different "places". And that what arises could also disintegrate. Nothing specific about world's re-arising.

If you are dealing with what Anaximander really thought, you are dealing with tiny scraps of course. Anaximander and the Origins of Greek Cosmology by Charles H. Kahn is a good standard reference.

 

[alt]

I've never seen it claimed that he meant a cyclic scenario for "our" world. Only that he made two claims - the apeiron was infinite and so would be able to produce an unlimited number of worlds in different "places". And that what arises could also disintegrate. Nothing specific about world's re-arising.

If you are dealing with what Anaximander really thought, you are dealing with tiny scraps of course. Anaximander and the Origins of Greek Cosmology by Charles H. Kahn is a good standard reference.

OK - thanks. That reference sounds interesting too. I'll have a look for it.

 

[ithinkther4im]

A thought occurred to me (about 10 years ago): I am not a physicist or a student of mathematics, just one of biology, however I am fascinated with the concept of the exxistence of God...as we all may be.



Try an experiment. Just try to prove the existence of 0. I had a room full of doctors and other professionals pulling their hair out on this one. If you divide 1 an infinite amount of times you will never get to 0. Also if I have 2 apples and I give you both, I have 0 apples but you then have 2, just a transfer of the matter, not a disappearance. 0 can be a placeholder as in the aforementioned equation, but not an actual entity.



If there is no such thing as zero, you need not a driving force to create it (God). Matter must be a self perpetuating force that we just cant understand because from our point of view, (Earth), everything lives and dies or gets destroyed. Even when the things get destroyed they are never really gone they are transformed into energy in the form of fire, fertilizer, radiation, etc. The point is we might not have the brain capacity to understand the universe yet, we are too small and insignificant. It would be like asking an ant to understand the internet. Sorry to be so long winded.....just a thought. All I know is the mystery lies in the Big Bang.

 

[Hurkyl]

Just try to prove the existence of 0.

From the Peano axioms, 0 is a natural number.
Therefore, there exists a natural number called 0. Q.E.D.

To keep you from further going off the deep end, I'm going to close this thread.