IS Math Real?

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IS Math Real?

Post by Hashi Lebwohl »

Here is a short video from Idea Channel about whether or not math is objectively real.

The basic premise is this: biology studies living things, physics studies motion and objects from particles to stars, chemistry studies the interactions of matter, etc. but mathematics studies...itself.

Also, did the concept of "3" exist before human beings came along--is "3" an objective facet of reality--or did we develop the idea of "3"? Bertrand Russell tried to answer questions like this.
We also need to ask other questions. If we have the idea of a triangle and what a cosine is, then would some extraterrestrial species--presuming they exist--arrive at the same idea, the ratio of the side adjacent to an angle to the hypotenuse? No, they won't use the word "cosine" but would they have the same idea? If so, then perhaps there is some sort of objective reality to mathematics and this is why Carl Sagan wrote it into Contact that aliens would use math for the foundation of communication. You may not be able to understand something I say but if I diagram the Pythagorean Theorem and your math is similiar to mine then we understand one another.

Anyway...interesting video. In fact the entire series is interesting and enjoyable.
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Post by wayfriend »

I have 2 apples. You give me 2 more. Now I have 4.

That's real.

The notion "2+2=4" is not someting you can point to when someone gives me apples. It's something that allows me to predict what will happen without actually having to have some apples.

So: Math is a mental discipline which is useful for predicting real things.

A mental discipline is objectively real. It's not apples, or people carrying them, or the act of giving them. But you can verify it exists objectively because I can say "4".
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Post by I'm Murrin »

I'd certainly say that accounting and geometry are fundamental. The rest are derived from those.

There may be many other ways of working with numbers, but they'll all be built on the same foundation.
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Post by Zarathustra »

We've done this before, but I always have fun with it.

When it comes to numbers, I'm somewhat of a Platonist. Yes, math/numbers/logical relations are objectively real, in the sense that their truths don't depend upon our subjective impressions. When I think "three," I'm thinking the same number you're thinking, even though each only occur "in our heads" in that moment. And any conclusion I can draw from a 3-sided geometrical figure, any intelligent/rational creature could also draw. Such conclusions aren't contingent upon biological facts, but upon logical necessity. We discover them, we don't create them. They precede us; they are always/already true.

Nor do they depend upon any correspondance with real objects in order to guarantee their truths. We don't have to map relationships of objects to relations between numbers to show that the numerical relationship holds. We use such comparisons the other way around, in fact: using math to confirm/explain/describe reality, rather than using objective reality to confirm math.

Even that move is problematic, in that we have no idea what justifies using math to describe reality; no idea why such a relationship between the Ideal and actual objects should hold.
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Post by Hashi Lebwohl »

I am in the same boat. Mathematics, as we uncover it, is a self-consistent system that accurately describes the world as we experience it. The ratio of the circumference of a circle to its diameter was a constant and existed before there were humans and will continue to exist as the same constant after the last of us dies. Given the fact that it would remain the same constant even if we all stopped believing in it proves that it must be objective reality--the physical laws of the universe dictate that this ratio is a constant.

Sure, this is probably old news for us but for the average person it is new because they have never thought of it before.

There is no "why" to explain why the relationship between the ideal and the actual holds true. We need only accept that it is true and proceed from there until proven wrong.
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Post by sgt.null »

maths is hard. that is real. so maths is real.
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Post by Avatar »

Of course it's not real. :D

It's purely imaginary. It's a contrived system intended to describe what we perceive. It may describe objectively exiting objects or phenomena, but that isn't the same as being real itself.

It works backwards, and that means that it can be anything it wants, as long as it describes the thing when worked forwards.

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Post by wayfriend »

Hashi Lebwohl wrote:I am in the same boat. Mathematics, as we uncover it, is a self-consistent system that accurately describes the world as we experience it. The ratio of the circumference of a circle to its diameter was a constant and existed before there were humans and will continue to exist as the same constant after the last of us dies. Given the fact that it would remain the same constant even if we all stopped believing in it proves that it must be objective reality--the physical laws of the universe dictate that this ratio is a constant.
That's not the proof you think it is. Because the notion of "constant" is a human observation, and you can argue that there's no such thing except in a human consciousness. Circles just are. They're not trying to be "constant". You have to satisfy whether the notion of "constant" is real.

No, you can't argue that Math, a mental notion, is real, by starting from a premise that another mental notion is real. Your just kicking the can down the road, as it were.
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Post by Hashi Lebwohl »

By "constant" I mean the things that existed before human beings existed to observe or think about them and the things that will continue to exist after we are all wormfood. The Sun is essentially the same as it was 1 billion years ago and it will be essentially the same 1 billion years from now, a time by which I suspect we will all be dead.
Still...there is truth in what you say. A circle is a human concept--the locus of all points equidistant from a given point--but even that includes the human concepts of "point" and "distance".

Would an extraterrestrial species have uncovered the same ratio of circumference/diameter regardless of the size of the circle in question? If so, then perhaps then we could say that "circles are constant" because we could rule out human psychology as a contributing factor to math being self-consistent. Unfortunately, we can never test that hypothesis until we make contact.

Just like with religion, we will only ever be able to conclude whether math is real to us on a personal level.
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Post by peter »

By 'constant', do we not mean here that there is nothing we could do to change it. No adoption of any different numbering system would ever alter the basic fact that the ratio of the cirrcumference of a circle to it's diameter is 3.141592654........ etc, etc of the units used to measure them. (Why is it this way any way - why not 3 or 4 or some other more rational number; wouldn't the Universe be simpler). If ever abducted by aliens this demonstation will be my first atempt at geting them to grasp that I am a sentient(ish) being!

Slightly at a tangent (no pun intended - well, yes actually ;) ) can I ask the suplementary question "Is math beautiful?" I may have mentioned in another post that I'm pretty much at the point where I believe our major contribution to the world - the one that 'raises us above the animals' [terrible way of putting it but you know what I mean] is our appreciation of beauty. This in my view is our only significant contribution to the whole - the fact that we can look out on our world and be stunned by it. We can also create beauty in our Arts. And this I thought was it. But then it occured to me that elegance in maths, in chess (related I think) is another form of beauty - albeit one closed to the untrained individual. In fact we could go further. Could the beauty of Math be the beauty that underlies all other forms of beauty - the rest being but shadows on the cave wall cast by the pure ideas of Math.

(Sorry Hashi if this is 'derailing your thread'; if it's too far from your original post, feel free to have it split off into a separate thread or discarded by the mods as they see fit.)
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Post by Hashi Lebwohl »

No, I much prefer any thread I start to have a life of its own without any mircomanaging on my part.

Yes, when I say "constant" in terms of math I am referring to something which will not change even if we all stop believing in it or will continue to be true even after we are all dead. If dogs develop more advanced sentience in the future, developing both language and the capacity for mathematics, they will also find that the ratio of circumference/diameter is the same regardless of the size of any circle they investigate.

I don't think there is a "why" that explains why pi is the number that it is without crossing the line over into theology. It merely is what it is. Interestingly, out of the uncountable transcendtal numbers that exist we know of only a small handful, usually special cases that arise from other mathematical investigations.

There is much in math that is definitely beautiful, especially various graphed functions or strange things like Klein bottles.
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Post by wayfriend »

peter wrote:By 'constant', do we not mean here that there is nothing we could do to change it.
It's a lot more than that. 'Constant' means that it doesn't change ... and it means that we consider this significant mathematically.
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Post by Zarathustra »

Hashi wrote:--the physical laws of the universe dictate that this ratio is a constant.
Hmm ... I don't think the physical laws have anything to do with it whatsoever. Geometry can be defined independently of physics or even of material reality. Pi would equal the same in any universe whatsoever [noneuclidean geometry could handle any kind of space] ... even in heaven. The truths of math are logically necessary, not contingent upon facts. This is precisely why the match between math and physical world are so inexplicable, because each operates on such different "laws." What happens in the future can never be known with mathematical certainty because events are contingent upon other events, and a dash of indeterminism. But the billionth decimal digit of pi will be the same no matter what.

This is also why math is always necessarily an approximation of the world. But that approximate relationship only makes the accuracy of our approximations all the more puzzling. If we lived in a clockwork universe where everything that happened was logically necessary and followed precise mathematical determinism, then it wouldn't be surprising in the slightest for there to exist a match between them. It would be tautological. But since events aren't necessary, much less logical, it is simply stunning to find this almost-symmetry between immaterial Ideals and messy facts.

There's your beauty right there, Peter. What could be more poignant and moving than an imperfect, deadly, uncaring world that "strives for perfection" inherent in math, even though the two "realms" are diametrically opposed, and yet still manages to get endearingly and enlighteningly close nonetheless?

Sometimes I can understand why some people would believe in god ... but it's always the opposite people you'd expect. Really, it's the scientists and mathematicians who see the true wonder of our reality. I'm not sure what everyone else is seeing. :lol:
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Post by Vraith »

why do people think pi is so special?
Look: the mystical value of pi never created one single thing, or solved one single thing. It is simply part of a definition. If pi DOESN't apply, it is because the thing isn't related to circularity. Pi is the result, not the determinant. Just like all things apple are not apple pi.
BECAUSE of circles Pi exists. Pi does not, cannot, cause circles.
[which is ugly to me in some ways, cuz one could say, because of watches, creation science exists...and that's just stupid.]

Which doesn't mean math isn't "real,"...it is simply real in a different way.
Just like words are.
And they have their limits.
Just like words do.
[and that is how math and words are both less real than people
and skyscrapers, but more real than God.
God has no limits, therefore God is false.]
And they lead to truth and tell lies.
A simple example is that every system of math that exists,
And, unless I'm badly mistaken
Every useful system that CAN exist
[don't fear folk, I'm not going to Godel here
which is the traditional next step]
depends
absolutely
on
undefined definitions
that must remain, forever, undefined.
depends on things that MUST be so
yet cannot, in any material sense
be so.
Lets not pretend Math is a pure race.
There are many peoples.
And, like the Star Trek flip-sided black/white/white/black people
they cannot touch without destroying each other.
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the hyperbole is a beauty...for we are then allowed to say a little more than the truth...and language is more efficient when it goes beyond reality than when it stops short of it.
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Post by Hashi Lebwohl »

Zarathustra wrote:Hmm ... I don't think the physical laws have anything to do with it whatsoever. Geometry can be defined independently of physics or even of material reality. Pi would equal the same in any universe whatsoever [noneuclidean geometry could handle any kind of space] ... even in heaven. The truths of math are logically necessary, not contingent upon facts. This is precisely why the match between math and physical world are so inexplicable, because each operates on such different "laws." What happens in the future can never be known with mathematical certainty because events are contingent upon other events, and a dash of indeterminism. But the billionth decimal digit of pi will be the same no matter what.

This is also why math is always necessarily an approximation of the world.
We will simply have to disagree here. Mathematics as we know it is based on observation of the physical world. Yes, it is at best only an approximation of the physical world but it is such an accurate approximation that it might as well be the physical world. Mathematics is the verisimilitude of the universe (I don't get to use that word very often, sadly)--only if the laws of physics change will mathematics as we know it fail.

Yes, Vraith, but pi shows up in places where circles do not exist, such as in {integral} e^(-x^2); look here under either "indefinite integral" or "definite integral". The natural logarithm, a function whose derivative equals the original function, gives rise to the ratio of circumference/diameter?
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Post by Ur Dead »

1 plus one sometimes does not equal 2.

1+1 is 11

11 equals 3

also

11 equals 17

depends how you add to and what base your using.
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Post by wayfriend »

Hashi,

Consider we define a square as right angles, equal sides. Then we define area of a rectangle as length times width.

When we calculate the area of a square ... lo, the area is equal to the square of the length of the side. For every square, every time. It's "constant".

But is it an artifact of the physical universe? Or is it just something that derives from how we define a square and how we define area?

Now we need to ask ... is pi an artifact of the physical universe? Or is it just something that derives from how we define a circle and how we define circumference?
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Post by Zarathustra »

Hashi, granted, math may have been invented in order to keep track of one's property, and it certainly has many uses, but we don't derive math from the world. Circles and squares are abstractions; they don't exist in the world. There are no points, and there are no lines. We invent entire branches of math that have nothing to do with physical reality. Sometimes we find that these invented mathematical curiosities have an unforeseen application to the world (like fractal geometry), so that the relationship you're implying is exactly backwards in many cases. Antimatter was discovered in much the same way, when the relevent equations had two solutions, one positive and one negative. It wasn't until positrons were discovered that we realized what this mathematical result meant.

However, the properties of a circle or square aren't merely a product of our definitions of circles and squares. They derive axiomatically from these definitions. "If you suppose a,b,c, then you must conclude x,y,z." The conclusions weren't built into the definitions consciously. Usually we discover them later, and they are often far from intuitive or expected. No one intended for the ratio of a diameter to a circumference to be an irrational number, much less that this ratio had applications well outside of the geometry of circles.

That's not at all like defining ordinary physical concepts, e.g. a "chair" as something to sit upon. You can sit on lots of things that aren't chairs, and you can make a chair that would be impossible to sit upon (one made of soap bubbles or CGI, for instance). There's nothing logically necessary about this defined object, and you can't say that any rational race would derive the same exact conclusions about it.

From Wikipedia on pure mathematics:
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics,[1] and at variance with the trend towards meeting the needs of navigation, astronomy, physics, engineering, and so on.

Another insightful view put forth is that pure mathematics is not necessarily applied mathematics: it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world.[2] Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing the real world, many applied mathematicians draw on tools and techniques which are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research.
From Wikipedia on The Unreasonable Effectiveness of Mathematics on the Natural Sciences:
Wigner begins his paper with the belief, common to all those familiar with mathematics, that mathematical concepts have applicability far beyond the context in which they were originally developed. Based on his experience, he says "it is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena." He then invokes the fundamental law of gravitation as an example. Originally used to model freely falling bodies on the surface of the earth, this law was extended on the basis of what Wigner terms "very scanty observations" to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations."

Another oft-cited example is Maxwell's equations, derived to model the elementary electrical and magnetic phenomena known as of the mid 19th century. These equations also describe radio waves, discovered by David Edward Hughes in 1879, around the time of James Clerk Maxwell's death. Wigner sums up his argument by saying that "the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it." He concludes his paper with the same question with which he began:

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
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Post by Orlion »

Yeah, I'm going to have to rebuke Vraith about his Pi rant ("Get thee behind me, Satan!" ;) )

Anyway, this flows nicely into Z's last quote: actual math tends to be applicable to more then what it was originally meant to describe. Take Pi: to the ancients and the 'simpleton' ( :twisted: :P ) it is a characteristic of a circle. That is too narrow a characteristic, it would be like describing a woman "merely as a blonde". Pi is actually a characteristic of cycles, and as a result is an integral part of trigonometry and wave mathematics. It's one reason why I :roll: every time that video that says Pi should be replaced with thi or something like that which is essentially 2*Pi because it is "more elegant". Such a suggestion only shows that the person who made it should have paid attention more in math class and just how "natural" Pi actually is.

Well... that leaves me to finally answer the OP. To me, the question of "Is math real?" is similar to asking "is English real?" They are man-made constructs that are used to represent what is actually out there... it is how we are able to study and interpret the natural universe around us... much how code converts to binary so the computer understands what is required of it.

So yes, if we ever come in contact with an alien race, certain mathematical oddities will be the first thing we are able to decode out of each other... namely shapes... and the periodic tables of the two species should be the same shape, even if there is no guarantee that the atomic weights will be the same.
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Post by Cozarkian »

peter wrote: I'm pretty much at the point where I believe our major contribution to the world - the one that 'raises us above the animals' [terrible way of putting it but you know what I mean] is our appreciation of beauty.
My dog appreciates beauty that I don't. Specifically, he appreciates the beauty of certain smells that my nose can't even distinguish. He even likes to roll in those smells so he can take the beauty with him.

As to the original post, I'm in agreement that math is real. I can verify 1 + 1 = 2 at least as easily as I can verify the existence of gravity. Just because the names of numbers and operators are man-given labels doesn't make it not real, any more than gravity wouldn't stop being real if we called it fudge instead.
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