What is Math? Where does it come from?

[sabianq]

Since everyone here is smarter than me, I was hoping that someone could help me figure this out.

to me, math is nothing more that a language that describes events and how the events may influence other events.

Is math nothing more than a way to describe a change "in" or a function "of"?
if "this" equals "this" and "that" equals "that" when "this" changes, then "this" must be a function of "that".

The relationship between mass and energy has always been there since the beginning of the universe, it wasn't until Einstein figured out how the correlation works and described it in symbols that we really understood the relationship.

In nature, there is no actual calculation of anything. but anything in nature can be described in a mathematical formula to show how the event works in real time. Like a river winding its way down to a sea. The River is following the path of least resistance which can be calculated.

so while maybe things in nature are do not actually calculate, Humans have derived a language to show how nature works. It seems to me that the only way we can describe nature in the form of a given mathematical set would be if nature followed certain basic laws that can be realized and described through a precise "language" that can be universally understood.

so mathematics is nothing more than a human construct to describe what we think we see in the world.

[cosmocrazy]

Math is a language, its used to measure an amount and predict an outcome. Math is longstanding, used well it is very accurate and above all truthful. Most scientists agree that math is probably the universal language of all intellectual life in the universe. where it came from, pass? probably as early as when we as a species started to measure (count) an amount?

[Jens]

so mathematics is nothing more than a human construct to describe what we think we see in the world.

It depends on how you look at it. According to Pythagoras, math is what the creator used to create the world. But in a sense, it is also a human construct. It's a bit like asking "what is language"? Language is what humans use to describe the world, but it is also a system of sounds uttered by people.

[mugaliens]

Since everyone here is smarter than me...

Are you sure? I thought you asked a very salient question!

To me, math is nothing more that a language that describes events and how the events may influence other events... ...so mathematics is nothing more than a human construct to describe what we think we see in the world.

I would agree that the physical world is the impetus for mankind's inventing of math, but that humans have gone further in math, able to describe things/relationships via math that don't, and won't ever exist.

I especially like the word "relationships," as that's what math does - it describes how things on one side of the equation relate to the things on the other side of the equation. For example, it describes how placing a weight in the center of a beam supported on both ends imparts compressive forces in the upper portion of the beam, tension in the lower portion, shear forces throughout, and a bending moment which remains constant throughout the beam, but changes it's sign at the center where the weight is placed.

On a more complex note, it also describes how heat varieous throughout a 3D object to which heat is applied to a portion of its surface, and it even incorporates time as one of the variables as the object doesn't heat up instantaneously. If you want to get into real rocket science, we can talk about compressible fluid flows as they pass through a rocket's nozzle.

Actually, we can't. While I know some basics, that math was a bit beyond me...

Above all, mathematics is a language, and a very precise one at that. I might be able to describe the rocket exhaust above in a thousand words, and while I would probably be able to convey those basic concepts, the exact values would be missing. If I could supply you with the equations, you could calculate all flows in all rocket nozzles using equations that contained less than a thousand characters, all told.

Assuming you understood the language, er, "the math."

[Neverfly]

Mathematics as a language is a human construct to describe.

But the actual performance of calculation in mathematics is not automatic. Since it is a description, no intelligence is needed for what is being ddescribed to perform, but it is needed to describe it.

[Cougar]

...Like a river winding its way down to a sea. The River is following the path of least resistance which can be calculated.

Just a quick interjection. I'm not sure if your "path of least resistance" quite covers the phenomenon. Rivers tend to meander around, which suggests something more than "least resistance" is going on. I think you'll want to look at this lesser-known paper by this young whippersnapper physicist, A. Einstein -- The Cause of the Formation of Meanders in the Courses of Rivers.

[Cougar]

"If the calculus comes to vibrant life in celestial mechanics, as it surely does, then this is evidence that the stars in the sheltering sky have a secret mathematical identity, an aspect of themselves that like some tremulous night flower they reveal only when the mathematician whispers." -- David Berlinski

[kleindoofy]

Math is a language, its used to measure an amount and predict an outcome. Math is longstanding, used well it is very accurate and above all truthful. Most scientists agree that math is probably the universal language of all intellectual life in the universe. ...

In this and the other above comments, one word is missing: axioms.

As I have always been taught, axioms are the fundamental, self-defining truths of our mathematical system, e.g. a = a, a + b = b + a, etc., which are accepted and used by all intelligent beings of whose existence we have knowledge (considering the nature of this forum, I'm stearing around saying "accepted universally"), and for which there is no deduction. As fundamental as they are, we all learn about them in elementary school, and subsequently proceed to forget about them.

The veracity of our mathematical conclusions stands and falls with the veracity of the axioms, separate from the validity of argumentation. This veracity isn't really a problem, since - as far as I know - our axioms are considered to be true as true can be.

Now, without venturing off into the dark and twisted woowoo channels of ATM, just as a thought, for some time now I've wondered if part of the solution in finding the Grand Unified Theory may lie in a re-evaluation of our axioms. Do they describe the truth of the human experience, only to lose their absolute standing in the ultimate conditions of the universe?

Being neither an astronomer, a physicist, nor a mathematician, that is just a thought spawned of idle curiosity. If need be, please let me down gently.

[mugaliens]

Just a quick interjection. I'm not sure if your "path of least resistance" quite covers the phenomenon. Rivers tend to meander around, which suggests something more than "least resistance" is going on. I think you'll want to look at this lesser-known paper by this young whippersnapper physicist, A. Einstein -- The Cause of the Formation of Meanders in the Courses of Rivers.

He did a fair job exploring the fluid dynamics, for a physicist.

However, he failed to explain why rivers meander. All he mentioned was that coriolis plays an effect on greater West-bank erosion in Southern-flowing rivers in the Northern hemisphere.

Although he hinted at it, Einstein failed to cover why rivers would still meander even if there were no planetary rotation at all.

[mugaliens]

Like a river winding its way down to a sea. The River is following the path of least resistance which can be calculated.

If you were to insert a water source on the North side of a perfectly flat (E-W) alluvial plane sloped downwards to the South, on a non-rotating planet (no coriolis) and come back after some time had elapsed, would your river had followed "the path of leaster resistance" and run in a straight line? Or would it have begun to meander back and forth until it looked like the Mississippi?

My hint begins with a capital M...

[Neverfly]

If you were to insert a water source on the North side of a perfectly flat (E-W) alluvial plane sloped downwards to the South, on a non-rotating planet (no coriolis) and come back after some time had elapsed, would your river had followed "the path of leaster resistance" and run in a straight line? Or would it have begun to meander back and forth until it looked like the Mississippi?

My hint begins with a capital M...

Um.. actually...

You've never played with a garden hose have you?

Yes, water flowing downhill can flow in a straight line.
It can also Meander.

Take a Metal Plate and run a tiny stream of water down it. It might meander all over the place. Surface Tension of the water is to blame as to why. The water bumps into its own edge.
If the metal plate is already wet, the water will go in a straight line. Surface tension was nullified by the plate being wet.
If you increase the flow of the water, it will straighten the line out too- The flow rate overwhelms the slight surface tension.
The surface tension reacts to the surface of the metal- even at the microscopic level.

Now lets go to a hill. You have dirt. Dirt that is not compacted in some areas and is in others. You also have variations in the texture of the hill, plants that block the way etc...

So now you have fluid mechanics, surface tension, Coriolis effect, rate of flow and still more factors to take into account.

Each factor has the ability to be the dominant factor depending on the element they are in and the state of the other factors.

I would agree with the original statement. Water will basically follow the path of least resistance going down a hill. Basically.
Because by that point, that's the dominant factor involved, even though you can point out other influential factors.

[William]

Mathematics requires a model which is a theoretical image of what will be studied. Mathematics is the term for the machinery and logic that is used to examine and analyze the models.

Numbers and formulae (A formula is a set of rules and formulated logic) without a model, has multiple meanings. For example, the number 42 can have many different meanings depending on the model it is associated with. (42 is taken from the joke in the Hitch Hikers guide to the galaxy. Question: What is the meaning of life? Answer: 42. An answer of 42 is meaningless with knowledge of the model 42 is connected with.)

There is the issue of translation error as the theoretical models are an imperfect image of the physical world. Mathematical calculations can be correct, while the model is not.

Comment:
1) As the physical world is different the current set of physics models, there is a translation error problem from the physical world to the models. The current set of physics models are inconsistent with each other (An example is so the called duality principle, light is both a particle and a wave).
2) The models are known to contain theoretical paradoxes. For example in Maxwell’s model a point charge requires an infinite amount of energy to hold it together which can not be correct as E=mc^2 and an electron has a finite mass, not an infinite mass. The question what holds “charge” together is not a sensible physical question, as charge is only a component in Maxwell’s model.
3) It is well known that something is fundamentally incorrect with Maxwell’s model. Electrons and protons are not static things (‘particles’) that carry properties, they are time varying entities.

[Ken G]

I would say that what math is, and where it comes from, is still a complete mystery. Certainly what it is is a process of applying logic to a set of axioms, and where it comes from is an intelligent mind, but somehow that just doesn't fully describe the situation that results. I would say that mathematics is more like a flower-- the axioms are the DNA in the seed, and logic is the soil and water, but the rays of sunlight that make it all work are not so well understood, as with any creative human effort. The flower grows into something different in important ways from seed and soil, and it's a mystery why. Are theorems created by a mind, or discovered in it? I think there's an emergent property of mathematics, and like most emergent properties, we have no idea why it emerges.

[Sam5]

If you were to insert a water source on the North side of a perfectly flat (E-W) alluvial plane sloped downwards to the South, on a non-rotating planet (no coriolis) and come back after some time had elapsed, would your river had followed "the path of leaster resistance" and run in a straight line? Or would it have begun to meander back and forth until it looked like the Mississippi?

...

Rivers alternate back and forth between cutting into the bank on one side of the river at one bend, and then cutting into the bank on the other side of the river at the next bend.

One of the main reasons a river meanders is because as the fastest side of a curving river (the side that cuts into a bank) swings around an outside loop at high speed, inertia causes the water to miss the next turn in the river, and the fast water coming out of the loop moves as straight as it can toward the opposite bank where it hits the bank on the other side of the river, which is the outside loop of that next bend.

The same left/right/left/right process is repeated all the way down river, in both the Northern and Southern Hemispheres and for East and West flowing rivers too.

http://waterknowledge.colostate.edu/meander.htm

http://geobytesgcse.blogspot.com/200...rs-ox-bow.html

See the alternating left/right pattern of sand bars in this photo:

http://www.uwsp.edu/geo/faculty/ritt..._GSC_large.jpg

[mugaliens]

Take a Metal Plate...

Actually, I believe I mentioned the term, "alluvial plain."

So now you have fluid mechanics, surface tension, Coriolis effect, rate of flow and still more factors to take into account.

Each factor has the ability to be the dominant factor depending on the element they are in and the state of the other factors.

I would agree with the original statement. Water will basically follow the path of least resistance going down a hill. Basically.
Because by that point, that's the dominant factor involved, even though you can point out other influential factors.

Not so. The key is turbulence. The "path of least resistance" for air flowing over a wing is rearward, yet a stall is the point at which laminar flow becomes turbulent.

Water flowing down an alluvial plain is alreay turbulent. The result is that it will very ever so slightly to the left, then back towards the path of least resistance.

The problem is that when it veered left, it began digging a channel in the silt, and that channel keeps getting deeper, until instead of just following the water towards the right, the path of least resistance, it overcurves, and begins channelling the water to, and beyond the path of least resistance. Once there, the water curves back to the left, again towards the path of least resistance.

Thus, what began as a straight shot, even in an almost completely homogenous medium, quickly becomes a series of meandering back and forth cuts.

Now...

Whether or not meandering occurs depends primarily on two factors - the angle of the slope and the erosion factor of the medium over/through which the water is travelling. If it's steep enough, the water will cut right through a potential bank, never allowing it to form. If the slope is gentler, the water doesn't have enough force to overpower the developing banks and meandering occurs.

[mugaliens]

One of the main reasons a river meanders is because as the fastest side of a curving river (the side that cuts into a bank) swings around an outside loop at high speed, inertia causes the water to miss the next turn in the river, and the fast water coming out of the loop moves as straight as it can toward the opposite bank where it hits the bank on the other side of the river, which is the outside loop of that next bend.

The same left/right/left/right process is repeated all the way down river, in both the Northern and Southern Hemispheres and for East and West flowing rivers too.

Bingo! You've got it, and thanks for the links!

[Ken G]

I would expect the key issue for a meandering river is called "instability", which asks, if you take a straight segment of river and give it a slight curve at random, does the slight curve tend to increase, or does it tend to straighten itself out? I would think the tendency to cut into the bank on the outside of the curve is the cause of instability. Eventually, the meander magnifies itself to a scale where the instability is no longer active, it "saturates in the nonlinear regime". The saturation physics is probably very complicated, and I'll bet that's where issues like steepness and softness of the banks come into play. Maybe if the surface soil is soft but the clay below is harder, once the stream cuts into the clay, the shape can change only much more slowly. Or maybe the shape is constantly in motion, such that a long-term time-lapse movie would look like a moving snake. Issues like that are raised in Sam5's links.

[kleindoofy]

... The "path of least resistance" for air flowing over a wing is rearward ...

Based on my reading on BAUT, the folks here tend to be rather exact, so please excuse my nitpicking:

In normal flight, a wing passes through unsuspecting, relatively calm air (that was floating around, minding its own business). This air is caused to move up or down around the shape of the wing passing through it and, as I suspect, doesn't really 'flow.' It does however receive movement energy from the wing, resulting in (part of) its subsequent movements, in addition to the differences in relative pressure and their rush to be equalized.

In a wind tunnel, it's the other way around, i.e. the air is moving and flows around a stationary wing.

Does that make any difference in the results?

[Andrew Barton]

In this and the other above comments, one word is missing: axioms.

As I have always been taught, axioms are the fundamental, self-defining truths of our mathematical system, e.g. a = a, a + b = b + a, etc., which are accepted and used by all intelligent beings of whose existence we have knowledge (considering the nature of this forum, I'm stearing around saying "accepted universally"), and for which there is no deduction. As fundamental as they are, we all learn about them in elementary school, and subsequently proceed to forget about them.

The veracity of our mathematical conclusions stands and falls with the veracity of the axioms, separate from the validity of argumentation. This veracity isn't really a problem, since - as far as I know - our axioms are considered to be true as true can be.

The idea that an axiom is a 'self-evident' truth has a very long and respectable history, but it's no longer the way that professionals think of it.

We often choose different sets of axioms to work with. One well-known example is the set of axioms Euclid chose for his geometry. You can replace his 'parallel postulate' axiom with other axioms and get equally consistent geometries. There is no way to tell by abstract reasoning which of these best describes the real world - you can only do so by observation and experiment.

Andrew

[Ken G]

In a wind tunnel, it's the other way around, i.e. the air is moving and flows around a stationary wing.

Does that make any difference in the results?

If it did, wouldn't it be unsettling that wind tunnels are used to design the wings on the planes in which we fly?

[Ken G]

We often choose different sets of axioms to work with. One well-known example is the set of axioms Euclid chose for his geometry. You can replace his 'parallel postulate' axiom with other axioms and get equally consistent geometries. There is no way to tell by abstract reasoning which of these best describes the real world - you can only do so by observation and experiment.

That's very true indeed. In fact, some mathematicians, like Hardy, reveled in the absence of any real-world applications. (As I recall, Hardy was very suspicious of how human power structures capitalized on mathematical results, so he prefered abstract results that did not connect with reality at all. Ironically, that turns out to be hard to do, and many of Hardy's own "abstract" results ended up having applications! That's also what I meant by the mystery of these "emergent aspects" of mathematics.)

[kleindoofy]

Mathematics requires a model which is a theoretical image of what will be studied. ... There is the issue of translation error as the theoretical models are an imperfect image of the physical world. ... The current set of physics models are inconsistent with each other ... The models are known to contain theoretical paradoxes. ...

... We often choose different sets of axioms to work with. ... There is no way to tell by abstract reasoning which of these best describes the real world ...

... In fact, some mathematicians, like Hardy, reveled in the absence of any real-world applications. ...

Ahh. Thank you. News to me.

How far does that go?

Can an axiom be irrational? Is it feasible (without getting caught up in the gooppidygook of ATM) that a set/sets of (hitherto unknown) axions/models would allow a solution to GUT? To bring certainty into Heisenberg's principle? To let us know just when Schroedinger's unfortunate feline goes to that giant litter box in the sky? To know what was happening during those furtive few minutes after the Big Bang? To rationalize Pi?

Just as RT augmented Newton for the fuzzy parts, can different axioms help us to understand the extremes?

[kleindoofy]

If it did, wouldn't it be unsettling that wind tunnels are used to design the wings on the planes in which we fly?

Well, yes. That was behind my question.

Possible answers might be / might have been:

-- Yes, there is a difference, but empirical data has shown it to be negligible, at least for a 300 metric ton airplane.

-- Yes, but that is compensated for by making mathematical adjustments to the results.

-- No, no difference.

[Ken G]

Just as RT augmented Newton for the fuzzy parts, can different axioms help us to understand the extremes?

I would say that's just what they do. Take quantum mechanics-- the axioms of quantum mechanics associate measurables with mathematical operators that act on spaces with imaginary elements that are "hidden" in the final answers that we can compare to experiment. I'd say that axiomatization is as about as "extreme" as you can imagine-- yet it works spectacularly.

[Ken G]

Possible answers might be / might have been:

-- Yes, there is a difference, but empirical data has shown it to be negligible, at least for a 300 metric ton airplane.

-- Yes, but that is compensated for by making mathematical adjustments to the results.

-- No, no difference.

The only difference I can think of is that the wind tunnel has boundaries that are closer to the wing than anything in real life. The fact that the air is moving, instead of the plane, is all a matter of perspective, and cannot matter at all except for how it interacts with the walls of the wind tunnel. I'm sure they do their best to keep such influences to a minimum, but only an airplane designer could tell you what precautions they need to take.

[Cougar]

In this and the other above comments, one word is missing: axioms.

As I have always been taught, axioms are the fundamental, self-defining truths of our mathematical system, e.g. a = a, a + b = b + a, etc., which are accepted and used by all intelligent beings of whose existence we have knowledge.... The veracity of our mathematical conclusions stands and falls with the veracity of the axioms, separate from the validity of argumentation. This veracity isn't really a problem, since - as far as I know - our axioms are considered to be true as true can be.

The idea that an axiom is a 'self-evident' truth has a very long and respectable history, but it's no longer the way that professionals think of it.

We often choose different sets of axioms to work with. One well-known example is the set of axioms Euclid chose for his geometry. You can replace his 'parallel postulate' axiom with other axioms and get equally consistent geometries. There is no way to tell by abstract reasoning which of these best describes the real world - you can only do so by observation and experiment.

Well, yes, it's a shift in level that results from supposing the negation of one of your former axioms. What if parallel lines DO cross? I doubt we'd find this so notable if it hadn't spawned numerous new geometries that are not only imaginative but also better adapted to describe particular situations.

Considering whether a+b always equals b+a -- a cinch for a self-evident truth -- yielded non-commutative algebras that turn out to be necessities in describing things quantum.

We often choose different sets of axioms to work with...

Which I don't think should be misunderstood as anything to do with postmodernism. Particularly, any one axiom or set of them does not have equal validity as any other. Many may lead to inconsistencies, and other sets may lead to systems that are hugely useful. As you say, such sets are subject to experimentation to see where the chips fall.

[Ken G]

Considering whether a+b always equals b+a -- a cinch for a self-evident truth -- yielded non-commutative algebras that turn out to be necessities in describing things quantum.

Maybe another way to say that is that a+b = b+a is not necessarily "self evident", because it really depends on the situation (addition might mean do both things, and if a=open a door and b=walk in, the order is going to matter). So the important thing math can do is be tailored to whatever you need it for-- any situation that involves the use of logic and consistency. Why the universe accepts that kind of treatment is the real big mystery-- did we develop logic to reflect a property of the universe, or does a universe have to be logical? If it doesn't have to be, then why is it? That's why I'm always more surprised when people expect life to be mathematically precise than when they don't.

[KLIK]

Just wanted to say re. rivers, that the average length of the average river is pi x the length as the crow flies which impresses me as a maths relationship to nature.

http://www.joyofpi.com/pifacts3.html (4kb) 4th item. simple description.

[Ken G]

Just wanted to say re. rivers, that the average length of the average river is pi x the length as the crow flies which impresses me as a maths relationship to nature.

That's interesting, note it would certainly be true if every meander of the river was a half-circle. That probably stems from the "nonlinear saturation" of the instability-- if a stream tries to meander more than a half circle it will have to double back on itself. On average, perhaps this "doubling back" point is where the meandering saturates, as it requires water to flow uphill.

[John Mendenhall]

Just wanted to say re. rivers, that the average length of the average river is pi x the length as the crow flies which impresses me as a maths relationship to nature.

http://www.joyofpi.com/pifacts3.html (4kb) 4th item. simple description.

Wonderful piece of information, didn't know that! Tjhank you.

Less seriously, and the length of the average BAUT thread as the Neverfly meanders?

Regards, John M.