adysav

I would say that maths is the only objective approach to reality.
This wouldn't be a sphere.
You can mathematically describe any object, but simplifcations are often used because.. they're simpler.

ARTelevision

anti fishstick, my response would be that to have an a priori notion of what is nature contradicts my statement "I have no idea what is really "out there"...

The acts of "seeing" or "observing" or "measuring" such things as golden rectangles or the golden ratio must involve the potentially erroneous practice of overlaying our own mental constructions on what is "out there."

That's the epistemological problem. The very act of interpreting golden-ratio-type relationships as preexisting in nature is a mathematical operation. What we do, it seems to me, is impose our internally consistent relationships upon whatever may be "out there." I can't see a way we could have any necessarily true idea of what nature is.

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shakran

OK, what about dogs, such as mine, that have not been trained to count, but if you mess up and give them the wrong number of treats, they bitch at you until you add in the missing ones.

They've not been trained to count, yet they know about quantity.

adysav

Sorry for jumping back, but I thought this needed addressing.
"Special" numbers like Pi and Phi are usually determined by the ratio of two quantities. The units you use to describe these quantities is irrelevant, as the result is always the same.
Same with the description of a sphere that someone mentioned.

Your statement is a bit like saying "the distance between New York and Tokyo is greater if you measure it in km rather than miles."

Master_Shake

Of course it would be a sphere! As a three-dimensional object the sphere would have to be constructed of something. Carbon atoms are just as good as anything else.

And yes, we use mathematics to describe objects, but the mathematics are irrelevant to the object. These simplifications are simple for us, not intrinsically. From a distance the Earth resembles a sphere (or an Oblate Spheroid if you prefer). For its inhabitants, most of the time, this macro-shape of the Earth is irrelevant. To us it is a collection of valleys, mountains, composite parts and different elements. For us to treat it as a sphere would often be useless. It is simpler to treat it as a bunch of component parts.

Consider, where do you live?

Apt. 2B, 1313 Mockingbird Lane, Scranton, Pennsylvania, United States of America, North America, Northern Hemisphere, Planet Earth, Sol System, Milky Way, etc... All of these answers are accurate, but only one is a meaningful answer to the question depending on who is asking and for what purpose. None of them are a universal truth, they all depend on the relationship of the person asking the question.

Try using mathematics to answer that question. You will still have to come up with arbitrary marks for a person to understand.

12 feet from the front door, or 600 Kilometers from the Washington Monument, or 42 degrees North Latitude, whatever. It's all dependent on the parties involved. There is no universal truth there, only a descriptive method for use between parties that have some frame of reference.

There's no reason dogs can't learn to count. But of course, we taught it to count, didn't we? It didn't discover it on its own. On its own, your dog probably wouldn't know it wants 5 milkbones, rather it would want enough milkbones to fill its stomach. It's the total mass of milkbone that matters to the dog, not the number of treats. Consider predatory animals in nature. They don't care if they take down 1 or 2 gazelle to feed. If the first gazelle is plump and its stomach is full, then its done for the day. But if the total amount of gazelle-age the animal has consumed is not enough to satisfy itself then it will take down a second gazelle. Not because it cares about the number it eats, but rather because it needs more stuff.

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You know something, I don't think the sun even... exists... in this place. 'Cause I've been up for hours, and hours, and hours, and the night never ends here.

antisuck

A sphere is an abstraction though, an ideal representation of a shape to which many things are similar, but nothing really is. A collection of carbon atoms in the approximate shape of a sphere is just that, a collection of things in an approximate shape.

A clearer example might be a Line, which clearly cannot exist having zero width, zero depth and infinite length. It's a useful abstraction though, a helpful model to have around. A sphere is a similar type of creature.

Master_Shake

I think saying nothing really <i>is</i> may be going to far. There probably is something, even if I don't know what it is. I can perceive parts of it and I label these perceptions.

I dub thee: sphere!

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You know something, I don't think the sun even... exists... in this place. 'Cause I've been up for hours, and hours, and hours, and the night never ends here.

shakran

That would be a good point, except that it's not right.

My dogs get all their treats at once. If I put down 4 (they get five) they won't touch 'em till I put the 5th one down there. Doesn't matter if I put 4 down fanned out, stacked up, or any other way, so it's not that they're seeing the wrong pattern on the floor either. They're counting 'em, and noticing that they're being shorted.


Also, we're talking about 3 basset hounds here. Basset hounds NEVER consider their stomach to be full, so if your argument were true for my dogs, they'd never stop bitching at me no matter how many treats I stuck down there.

Master_Shake

Dude, there's no reason dogs can't learn to count. When you're dealing with domesticated animals that are fed sufficiently similar items on a regular basis they become conditioned to eating that many milkbones. I'm sure the dogs would love more milkbones, but they've become conditioned to accept the fact that you're only going to give them 5 milkbones. Try giving them 6 milkbones for a month, then try going back to 5 (or whatever the number is).

Without your training, the dogs would almost certainly eat until they were full (or some other cause stopped them). Where the animal behaviourists at? Can anybody back me up that animals don't much care for numbers on their own?

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You know something, I don't think the sun even... exists... in this place. 'Cause I've been up for hours, and hours, and hours, and the night never ends here.

Suave

asaris, math IS a science, and is also the basis of all the natural sciences.

stingc

Everyone seems stuck on very archaic concepts of mathematics. As a whole, it is not based on the concept of numbers, and is quite separate from requiring any type of physical reality.

I think it would be useful to describe how a modern mathematician works. They basically do two things. The first of these is to set out definitions. Assume the (abstract) existence of an object X satisfying logical properties (1), (2), ... Any object with these properties is given some arbitrary name. This process seems quite clearly to be invention.

Once everything has been defined, the mathematician then tries to find interesting (meaning nontrivial) relations between them. These properties are logically deduced using only the definitions already given.

It is a little harder to say whether this step should be called discovery or invention. One is given a set system, and is finding rules for it that weren't known before. This is very closely analogous to the physicist finding rules to describe the universe. Both systems obeyed those rules independently of the discoverer(/inventor)'s understanding. I would therefore say that this part of mathematics is discovery.

There is an obvious counter-argument though. In the case of mathematics, all theorems are logically contained in the definitions (which are invented). It may therefore be said that the theorems were invented at the same time as the definitions. The statement is then that the inventor was simply not smart enough to understand the depth of his creations. Although I can see the point of this argument, it seems to run against the most common usage of the words we're discussing.

So I think math is fundamentally about discovery, but not the same sort of discovery that the traditional sciences strive for. Scientists have but one universe to study. Mathematicians create their own, and produce a new one whenever they get bored. In this sense, math contains elements both of the arts and sciences.

More practically though (to those of us who would like to actually use math for something practical), it is best thought of as a highly developed form of logical argument.

ARTelevision

A close reading of this thread indicates that not everyone is stuck on very archaic concepts of mathematics.

Thanks for your views.

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molloby

The fact is that every single concept in mathematics is an abstraction, from unity upwards.

It is a frame-work that we put around the world in which we live in the same way that physics is. It is however a human construction and a product of how we percieve the world.

It would be extremely interesting to see what another intelligent race would come up with in its place. I see it as hubris, however, to assume that it would be much alike.

dbc

Math is discovered. The relationship between numbers is there. Someone just discovers those relationships, he/she doesn't invent them.

Karkaboosh1

language/tool

asaris

Mind you, I tend to be a constructiveness about nature, too. I've read too much Kant and Heidegger, I suppose. In a nutshell (and it'll sound loonier than it really is), all things are created by the human mind. There are no things "out there". There's no "there" out there.

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"The Germans believe that power must reveal itself in hardness and cruelty and then submit themselves gladly and with admiration[...]. They do not believe readily that there is power in meekness and calm."

-- Friedrich Nietzsche

daking

axioms are invention.
proofs are invention.
theorems are discovery.

I guess apart from pythagoras's theorem as its equivalent to Eucilds 5th postulate. So that would make it an invention if viewed in that perspective.

asaris

I wouldn't want to say that Math is a science, any more than I would want to say Philosophy is a science, and philosophy underlies science as much as math does. The experimental method seems to be fairly important to a natural science (though maybe I'm just being archaic again), and I've never known Math to use the experimental method. Of course, given what I've written above, it wouldn't hurt my point if math were a natural science.

And who's using an archaic notion of math? I'd be willing to bet that I've taken math classes at least as advanced as anyone else on this board; in fact, since I've taken a graduate level math course, it'd be hard for anyone to have taken a more advanced course. So you might want a little more content in that accusation.

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"Die Deutschen meinen, daß die Kraft sich in Härte und Grausamkeit offenbaren müsse, sie unterwerfen sich dann gerne und mit Bewunderung:[...]. Daß es Kraft giebt in der Milde und Stille, das glauben sie nicht leicht."

"The Germans believe that power must reveal itself in hardness and cruelty and then submit themselves gladly and with admiration[...]. They do not believe readily that there is power in meekness and calm."

-- Friedrich Nietzsche

filtherton

I agree, math is a convenient construction used to describe the world in which we live. I think math is an invented means to describe and make use of discovered things.

livingfossil

Simply because something is not known does not make it untrue. Simply because something cannot be understood does not make it untrue. The discovery of 'that tribe' (the piranhas, I think) demonstrates the inherent limitation of all languages. But it has little to do with math or its efficacy.

Now, whether math can aid us in understanding natural phenomena is indisputable. It can, and does. Whether I understand it, have words for it, or not, it will do the same. Naming something one, un, uno, or whatever has no bearing on the matter.

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kd4

yes

our numerical system is all made up. We as a society have decided that the symbol 1 represents one of said item and so forth. This includes language, math is an invention.

daking

Well that depends, a system can be inherently inconsistent. At which point it gets rejected by those who might use it.

For instance a number system where somehow 3+2=5 and 3+2=6 would be break down pretty quickly. For it to be consistent a whole new layer of construction and method would need to be created. Some kind of modulus arithmetic.

asaris

But kd4, could the quantities represented by those numbers be such that 3 + 2 = 6?

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"Die Deutschen meinen, daß die Kraft sich in Härte und Grausamkeit offenbaren müsse, sie unterwerfen sich dann gerne und mit Bewunderung:[...]. Daß es Kraft giebt in der Milde und Stille, das glauben sie nicht leicht."

"The Germans believe that power must reveal itself in hardness and cruelty and then submit themselves gladly and with admiration[...]. They do not believe readily that there is power in meekness and calm."

-- Friedrich Nietzsche

stingc

Math does use the experimental method in a particular sense. Imagine that you are learning some type of math, and have just been presented with a set of definitions. These will generally seem to be too abstract to grasp on any sort of intuitive level, so you generally want to construct some examples to get a feel for how things work. Say that you now have several different examples, but are surprised to notice that they all share one particular property. You might then guess that all possible examples share this propery. So your examples have now given you an idea for a possible theorem. Examples are even more powerful in showing that a particular property is not correct when intuition would imply the opposite.

A well-known example would be Fermat's Last Theorem. Whatever progress Fermat may have made in proving it (nobody knows), the rest of the world didn't have a proof until a few years ago. It was, however, thought to be true because it seemed to work for every example that was tried. People kept showing that more and more special cases of it were correct. This gave people enough motivation to continue to try finding a complete proof even after 300 years.

So "experimentation" with mathematical definitions is an important part of mathematical progress. Unlike in the natural sciences, however, it is not absolutely required. Also, unlike math, science can never provide absolute proofs of anything.

I spend my days doing mathematical physics, so I also have plenty of experience with math. Its not terribly relevant though. I overspoke by saying "everyone," and I apologize.

stingc

Sure. As long as other additions keep the system consistent. And you can define consistency however you like, so you can really do anything at all.

Modern math considers numbers as we've all learned about them to be a particular example of a much more general type of object. For example, the integers form a ring, and the real numbers are a type of field. If you follow those two links and some of the pages linked from them, you'll start to get an idea of the types of things that you could come up with if you wanted to. Of course, you could say that those definitions aren't general enough either, but then you're really better off not using + and * signs anymore if you want anybody to read what you're doing.

All of these concepts are supposed to be completely abstract. The fact that we commonly use numerical quantities (along with the standard addition and multiplication operations) in describing the physical world is a completely separate philosophical problem.

kd4

yea, would have said the exact same thing

joeshoe

I say basic math is discovery, because they have real-world meanings, like the number of objects and so on. Beyond calculus, I would say invention, because you start to define more and more concepts.

mtb_chris

Math is a construct we have invented to represent things that we see. It may, however, be able to fundamentally represent everything in nature: if energy and matter are quantum, then can't they be expressed fully in numbers (it is my understanding that we can't measure them, but if we could..)?

Here's a mind bender: can god (assume god exists) change the value of pi? or phi? or e? or 1:1?

1st post

fuzzybottom

in·ven·tion ( P ) Pronunciation Key (n-vnshn)
n.
The act or process of inventing: used a technique of her own invention.
A new device, method, or process developed from study and experimentation: the phonograph, an invention attributed to Thomas Edison.
A discovery; a finding.

i vew math as recognition of relationships and patterns in and between quantities. we can create the symbols and methods for interpreting and dealing with what we discover, and ways to apply what we find, but they were related before we found them. though, i could say 5x=12y² and invent a relationship that doesn't necessarily have any real counterpart. depends whether an idea is considered an invention, i guess.

apparently, its all the same anyway.

Reese

I have so rarely seen math proven wrong so therefore it cannot be man made.

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Xell101

Math is an ascription of terms and methodologies used to process information found in the human psyche to the universe. I view it as being an intellectual construct used for processing information that exists soley inside the human mind. Does math exist outside the human in the world, or in the human as a means of processing that world in understood terms? I believe that the universe is bigger than the human mind, therefore math is something found within the human mind that serves as a looking glass, as opposed to being something fundamental to the the world or universe that humans can percieve suffeciently well to have, or at least claim, profeciency with.

That's my "I haven't had good sleep for 164+ hours and haven't had a good dose of caffeine for 40+" answer.