[Verv]

I was wondering about what mathematics are and their role within our society, and what role in reality that they have. This is going to be sloppy, but remember, it is in part meant to create a greater discussion:

Numbers are ideas that we created to represent the enumeration of things, (a) such as objects, or such as other theoretical creations used for measuring.

An example of (a) is there being 5 cows in the pen, or 10 people in the office, or 48 logs of wood in the wood pile.

An example of (b) is there being 12 inches in a foot, and this person is precisely six feet tall. It is 28 miles to Dallas. It is the year 2015 on our calendar.

Math was a systematic understanding of the relationships of these numbers. Its most important uses are the most obvious ones -- adding or subtracting, multiplying, dividing. It can be used in a whole variety of different ways -- some practical, some theoretical.

It is practical in so far as it serves a purpose of direct relation to our own existence -- mostly in the understanding of the quantification of things around us, and the results of their increasing and decreasing.

We begin to define it in theoretical terms when it strays past the point of it being directly relevant to our lives. And when it becomes theoretical, it is largely operating off of that which we know ground entirely in the reality of the practical mathematics.

Theoretical mathematics is necessarily an extrapolation off of the practical -- if it strays beyond what can be concluded by the practical reality of mathematics, it cannot be known to have a truth value or not.

But what can be concluded from it?

I do not see anything beyond the practical uses that helps us directly.

We are told that mathematics is the language of the universe and other such platitudes, and it is certainly true that since time, distance, and even things such as words and sounds can have mathematics applied to them in a system of measures, in fact in some instances forced upon them, mathematics can be a relevant instrument in the understanding of phenomena....

But it isn't the language of the universe, nor is it an indicator of any larger truths. It is purely instrumental.

I'd love ot hear other people's thoughts on this, whether in response to me or whether completely on a different vein of understanding math.

Thank you.

[Harmattan]

I also disagree about them being the "language of the universe" but I think they are more than mere instruments. They are a representation, an embodiment, an incarnation of the universe.

They are not just utilitarian, they are an equivalence (lossless or quasi-lossless) and they have a worth of their own (without taking into account their benefits) just like art. They are a golem into which we trap the universe, just like photography embodies light and digital audio embodies music, although those two are far more lossless. By the way, note that light, music and almost everything have no existence of their own, they are mere perceptions - representations again - of the universe just like the universe itself may be the representation of something else.

Basically you could say that mathematics are as transcendental as any human concept might be from a human POV. And any advanced extraterrestrial civilization would without any doubt share many or most of our mathematical concepts ; on the opposite it is unlikely that they could find any interest in our musical or picture arts. The latter depend on our senses, not mathematics. Mathematics are possibly the only universal language, which proves in my opinion how well they embody the universe.

Theoretical mathematics is necessarily an extrapolation off of the practical -- if it strays beyond what can be concluded by the practical reality of mathematics, it cannot be known to have a truth value or not.

I strongly disagree with this one. Mathematics only need to accept a few axioms (two or three principles based on empirical observations) and they can derive anything from that - whole mountains of theories. And believe me that anytime those theories found some practical use, their validity was systematically verified. Mathematics are not an experimental science, they are at heart a theoretical corpse driven by theoretical considerations. Theory typically precedes practical needs.

By the way you are focused on calculability but mathematics in my experience are more about description. Calculability is just a nice side-effect that also makes maths useful.

[Lexington]

I think "language of the universe" is actually a good way to describe mathematics - it's an effective way of describing reality. It's all shorthand for counting things in the end; the language used to express it just gets more complicated.

Addition is just counting things; we count two things, then two more, we have four things. Subtraction, we count one less and we have three. Multiplication is how many things can I have if I have three groups of three things: nine. Division is how many things I can count into a group; out of ten things and I want to split it into 5 evenly, 2 apiece.

Exponentiation is multiplication unto itself: if I start with one thing, then have ten of them, then ten of that, it's 10^2: 100. The logarithm (base 10) is the reverse: given that I have 100 things, how many times do I multiply by 10 to get there? (2).

In algebra, one describes relationships between things. If you have a population increasing by 10% each year, f(t) = 1.1^t. This is still just counting, but we've just explained it in terms of time: at time t what is the population? At one point there are a hundred people; the next 110; the next 121.

Trigonometry is still about counting: if I draw a triangle in the sand or want to figure out how long a suspension cord should be, it's measuring how long the long side has to be.

Calculus is taking algebra one step up: how fast is the line going up or how much space is there beneath the line. You're still counting - one could (painfully) describe it in terms of counting but it's still shorthand for just adding and subtracting things at the end of the day, even if you're doing in infinitely small increments.

[Saeko]

I think mathematics is far far deeper than mere counting. I think mathematics is about taking the simplest idealized objects with certain properties and then looking at those transformations of those objects which leave said properties unchanged. When you study the interactions of these transformations, objects, and properties they seem to take on a life of their own.

To illustrate, we usually think of numbers as objects, as collections of things, as "nouns". So, for example, 2 + 3 = 5 can be interpreted as combining a collection of 2 units with a collection of 3 units to get a collection of five units. But we can also think of numbers as transformations, as "verbs". In this case, we can reinterpret the previous equation as taking the collection of 2 units (the object) and applying the transformation "add 3 units" to obtain a new object, 5 units. More abstractly still, we can think of both of the addends as transformations on an arbitrary object, and their sum is a new transformation. Then the equation says, "take any collection whatsoever, then adding 2 units and then adding 3 units is the same as adding 5 units".

Here comes the fun part. Often, there are more transformations between objects than there are objects, and we can treat transformations themselves as objects and thus create entirely new objects. If we consider the transformation which leaves any collection alone, we can call it the "+0" transformation, and thus create the new object "0". We can further observe that any transformation such as "add x" can be reversed, so that we can get a new transformation "remove x", and then we get the negative whole numbers.

Conceiving of numbers as transformation and using a little set theory, we can prove that the associative law is true for integers, and then from that follows the commutative law, and, ultimately, we get a special kind of mathematical structure over the integers called an Abelian Group. An Abelian Group has the following laws:

a + (b + c) = (a + b) + c (associativity)

a + b = b + a (commutativity)

a + 0 = 0 + a = a (identity)

a + (-a) = 0 (inverse).

Now comes the really cool part. If we take these laws as our given structure and consider transformations of the entire set of integers which leave this structure invariant, we get all of the laws of multiplication and all the rational numbers for free! To do this, all we have to do is consider transformations of numbers which map the sum of any two numbers unto the sum of the numbers which the old numbers are mapped to. That is, if we have a + b = c, and we change a into x, and b into y, and c into z, then the sum of x and y must be z, i.e., x + y = z. Or, writing this in function notation, a + b = c -> f(a) + f(b) = f(c).

The best part is that we don't have to painstakingly change every single number one by one. All we have to know is what the transformation does to a single number, and we get all the rest for free.

So, if say, the transformation takes 2 -> 6, then it must take 5 -> 15, 13 -> 39, and so on. To see why, consider the equations 1 + 1 = 2 and the equation it maps to is f(1) + f(1) = 6. There is only one number which, when added to itself gets you 6, and that number is 3, so f(1) = 3, and we see that this transformation is just multiplying by 3.

Any transformation such as f(a) = b, defines uniquely the rational number b/a. Multiplying two rational numbers together is then defined by composing their respective transformations together. So, for example, if we have f(1) = 7, and g(5) = 3, then g(f(2)) = (3/5)*7*2 = 42/5 as you can check. And from here it's only a few easy steps to prove all of the familiar properties of multiplication, including the distributive property. This is why multiplication obeys many of the same laws as addition despite the fact that they are so different conceptually. Multiplication is what you get when you consider the interactions between transformations of numbers which leave the properties of addition unchanged.

What results is a mathematical structure called a "field", and the mystery deepens here because you can't take structure preserving transformations of the field into itself to get a new structure like we did above with a group. In some sense, the field already has "as much structure as possible" and there simply isn't much room for more.

I would also recommend the book A mathematical gift I: the interplay between topology, functions, geometry, and algebra (which is available for free online), which is accessible to anyone and hopefully conveys the feeling of higher mathematics with very neat ideas from topology.

[lucky]

Edit: nevermind.

[RedPillAger]

i view mathematics as a language, one that is extremely adept at describing our universe. what could take many pages of text to describe, can sometimes be concentrated into a single equation.

[Jim4120]

I like Saeko's description. This is why I enjoy math so much, and what makes number theory such an interesting topic to study. Math can be used as a wonderfully effective way of describing the universe around us, hence its use in science, especially the fields of physics, chemistry, and biology. It can even be used to study itself, which is interesting to me as well.

When you study the interactions of these transformations, objects, and properties they seem to take on a life of their own.

The only thing I would say is that I wonder what this means. Does it mean that math is some universal truth, or is it just a result of our excellent ability to see patterns in things that we make observances such as this? Said in another way, do the emergent properties of math have some deeper meaning, or is it really just us seeing something meaningful when there isn't really anything meaningful there?

[Ummon]

I do not entirely agree with this view, but I think it is important that it be presented:

http://www.scientificamerican.com/artic ... h-excerpt/

I feel like feynman that some people make fundamental attribution errors when thinking about these things (ie they mistake the map for the territory). They confuse mathematical objects with real objective things which often can be the case, but often is not.

“Physics is to math what sex is to masturbation.”
― Richard P. Feynman

With that said I have the deepest respect for men like Erdos, Grothendieck, etc and I think and feel that such "mathematical mystics" are often geniuses of the highest order.

[Ummon]

To answer the question properly though, I think math is the closest thing we have to a way of objectively relating experiences to one another. Even for those who are not good with equations a symphony can make you cry, or the motion of dance can send a message, poetry van spur insight. For me all of these expressions are transformations of one another and convey something deeper. It is the way we express meaning and value. So for me math is a metaphor. I think Herman Hesse attempted to convey something similar in The Glass Bead Game. I think that physics is starting to see these interstices between seemingly unrelated things by means of "information." We are beginning to see the same in mathematics as well: http://phys.org/news/2014-12-xinwen-zhu ... atics.html

All of these theories of everything, relationships between structure and function, etc are just ways of integrating our experience in a precise language that can convey to another what you're trying to express, however sometimes I find it lacking if the think is a proficient who has "lost the music."

[UnusuallyUsual]

I would say that I am a Dualist and I believe there is a Physical aspect to Reality and then *also* a Logico-Mathematical aspect to Reality. "Somehow" ( I'm not really sure how to describe or explain this ) sufficiently complicated physical conglomerations of matter called "brains" are able to perceive the Logico-Mathematical aspect. However, and I cannot stress this strongly enough, the two aspects are NOT able to interact in any way. In other words, people can never "do" anything regarding math, simply understand more of it. I would also say one of the most fundamental distinctions between these two aspects of Reality is Finitude. All of the Physical aspect is Finite, and all of the Logico-Mathematical aspect is infinite.

[yiostheoy]

I think mathematics is far far deeper than mere counting. I think mathematics is about taking the simplest idealized objects with certain properties and then looking at those transformations of those objects which leave said properties unchanged. When you study the interactions of these transformations, objects, and properties they seem to take on a life of their own.

To illustrate, we usually think of numbers as objects, as collections of things, as "nouns". So, for example, 2 + 3 = 5 can be interpreted as combining a collection of 2 units with a collection of 3 units to get a collection of five units. But we can also think of numbers as transformations, as "verbs". In this case, we can reinterpret the previous equation as taking the collection of 2 units (the object) and applying the transformation "add 3 units" to obtain a new object, 5 units. More abstractly still, we can think of both of the addends as transformations on an arbitrary object, and their sum is a new transformation. Then the equation says, "take any collection whatsoever, then adding 2 units and then adding 3 units is the same as adding 5 units".

Here comes the fun part. Often, there are more transformations between objects than there are objects, and we can treat transformations themselves as objects and thus create entirely new objects. If we consider the transformation which leaves any collection alone, we can call it the "+0" transformation, and thus create the new object "0". We can further observe that any transformation such as "add x" can be reversed, so that we can get a new transformation "remove x", and then we get the negative whole numbers.

Conceiving of numbers as transformation and using a little set theory, we can prove that the associative law is true for integers, and then from that follows the commutative law, and, ultimately, we get a special kind of mathematical structure over the integers called an Abelian Group. An Abelian Group has the following laws:

a + (b + c) = (a + b) + c (associativity)

a + b = b + a (commutativity)

a + 0 = 0 + a = a (identity)

a + (-a) = 0 (inverse).

Now comes the really cool part. If we take these laws as our given structure and consider transformations of the entire set of integers which leave this structure invariant, we get all of the laws of multiplication and all the rational numbers for free! To do this, all we have to do is consider transformations of numbers which map the sum of any two numbers unto the sum of the numbers which the old numbers are mapped to. That is, if we have a + b = c, and we change a into x, and b into y, and c into z, then the sum of x and y must be z, i.e., x + y = z. Or, writing this in function notation, a + b = c -> f(a) + f(b) = f(c).

The best part is that we don't have to painstakingly change every single number one by one. All we have to know is what the transformation does to a single number, and we get all the rest for free.

So, if say, the transformation takes 2 -> 6, then it must take 5 -> 15, 13 -> 39, and so on. To see why, consider the equations 1 + 1 = 2 and the equation it maps to is f(1) + f(1) = 6. There is only one number which, when added to itself gets you 6, and that number is 3, so f(1) = 3, and we see that this transformation is just multiplying by 3.

Any transformation such as f(a) = b, defines uniquely the rational number b/a. Multiplying two rational numbers together is then defined by composing their respective transformations together. So, for example, if we have f(1) = 7, and g(5) = 3, then g(f(2)) = (3/5)*7*2 = 42/5 as you can check. And from here it's only a few easy steps to prove all of the familiar properties of multiplication, including the distributive property. This is why multiplication obeys many of the same laws as addition despite the fact that they are so different conceptually. Multiplication is what you get when you consider the interactions between transformations of numbers which leave the properties of addition unchanged.

What results is a mathematical structure called a "field", and the mystery deepens here because you can't take structure preserving transformations of the field into itself to get a new structure like we did above with a group. In some sense, the field already has "as much structure as possible" and there simply isn't much room for more.

I would also recommend the book A mathematical gift I: the interplay between topology, functions, geometry, and algebra (which is available for free online), which is accessible to anyone and hopefully conveys the feeling of higher mathematics with very neat ideas from topology.

Saeko has nicely summarized what I call "pure math".

This is where/when you create your mathematical system with certain definitions, and then work those operations from those definitions to see what you get. Ultimately you can then go deeper and deeper into the theoretical nature of it all -- such as with multi dimensional integral calculus.

THEN in addition there is practical math or applied math. This is where we have created (yes, we created math, it does not exist on its own, sorry) our set of definitions and operations, and then we find things in the Universe that behave just like our mathematical models.

The thing to remember always is that the model is not the phenomenon. The model is only an approximation for the phenomenon which helps us to comprehend the phenomenon. Most scientists make this mistake in science as well as in math. Then they get so carried away in their theories that they get lost in some corner of the mind which has no application to the real Universe at all.

Aristotle was guilty of this with his crystal spheres. He completely overlooked the forces of gravity. There were never any crystal spheres.

Stephen Hawking makes the same mistakes today as well with his astrophysical models of quirks and quarks.

Getting back to math, and using Occam's Razor on it, we created it (yes we did!) for counting things -- antelopes out in the savannah -- lions in a competing hunting pride. Then we invented Geometry to help us build strong beautiful structures. Eventually differential equations and calculus came along. Now the things we can build and the satellites and planets we can aim our rockets at have become immense in their scale.

But like the Tooth Faeire and Santa Claus, math is just an invention of the human mind.

Once humans cease to be, if that day ever comes, then math will cease to exist as well.

[yiostheoy]

I was wondering about what mathematics are and their role within our society, and what role in reality that they have. This is going to be sloppy, but remember, it is in part meant to create a greater discussion:

Numbers are ideas that we created to represent the enumeration of things, (a) such as objects, or such as other theoretical creations used for measuring.

An example of (a) is there being 5 cows in the pen, or 10 people in the office, or 48 logs of wood in the wood pile.

An example of (b) is there being 12 inches in a foot, and this person is precisely six feet tall. It is 28 miles to Dallas. It is the year 2015 on our calendar.

Math was a systematic understanding of the relationships of these numbers. Its most important uses are the most obvious ones -- adding or subtracting, multiplying, dividing. It can be used in a whole variety of different ways -- some practical, some theoretical.

It is practical in so far as it serves a purpose of direct relation to our own existence -- mostly in the understanding of the quantification of things around us, and the results of their increasing and decreasing.

We begin to define it in theoretical terms when it strays past the point of it being directly relevant to our lives. And when it becomes theoretical, it is largely operating off of that which we know ground entirely in the reality of the practical mathematics.

Theoretical mathematics is necessarily an extrapolation off of the practical -- if it strays beyond what can be concluded by the practical reality of mathematics, it cannot be known to have a truth value or not.

But what can be concluded from it?

I do not see anything beyond the practical uses that helps us directly.

We are told that mathematics is the language of the universe and other such platitudes, and it is certainly true that since time, distance, and even things such as words and sounds can have mathematics applied to them in a system of measures, in fact in some instances forced upon them, mathematics can be a relevant instrument in the understanding of phenomena....

But it isn't the language of the universe, nor is it an indicator of any larger truths. It is purely instrumental.

I'd love ot hear other people's thoughts on this, whether in response to me or whether completely on a different vein of understanding math.

Thank you.

I was going to quote you, however I had to address Saeko's formulation of "pure math" first or else she could have easily rebutted me.

[Paradigm]

Math is a tracing of the real. It is reality abstracted from the continuous flow of time. When you move your arm in a continuous motion, the points through which your arm passes can be plotted on a graph and described through an equation. The confusion comes when we then try to say that the motion itself was that of your arm moving from one point to another, when in fact the motion was continuous while the mathematics describing it are a tracing. Every one of Zeno's paradoxes is a confusion of this type. Even with simple numbers, we err when we think we are measuring quantity, when in fact quantity is that by which things are measured. What mathematics cannot describe is the spontaneous flow of time, though when it does try to describe time, it is again describing a tracing of it.

[yiostheoy]

Math is a tracing of the real. It is reality abstracted from the continuous flow of time. When you move your arm in a continuous motion, the points through which your arm passes can be plotted on a graph and described through an equation. The confusion comes when we then try to say that the motion itself was that of your arm moving from one point to another, when in fact the motion was continuous while the mathematics describing it are a tracing. Every one of Zeno's paradoxes is a confusion of this type. Even with simple numbers, we err when we think we are measuring quantity, when in fact quantity is that by which things are measured. What mathematics cannot describe is the spontaneous flow of time, though when it does try to describe time, it is again describing a tracing of it.

... Except that the "flow of time" does not exist either -- we just think and imagine that it does because the Earth rotates on its axis in the proximity of the Sun creating the illusion of what we call "days" and "nights" which seem to follow each other and thus somehow mark out what we call "time".

But time does not really exist. Same as math, which also does not really exist. They are both just all in our heads.

[Hong Wu]

Math is manipulating symbols which correspond, usually imperfectly, with reality. We don't notice though because if we want two cups, and there are two things which meet that definition, the imprecision doesn't impact us. For example, one cup might contain more material than another but what we really want is that which qualifies to our concept of "cup" and things like the number of molecules in each cup might not matter.

[yiostheoy]

Math is manipulating symbols which correspond, usually imperfectly, with reality. We don't notice though because if we want two cups, and there are two things which meet that definition, the imprecision doesn't impact us. For example, one cup might contain more material than another but what we really want is that which qualifies to our concept of "cup" and things like the number of molecules in each cup might not matter.

That it is indeed, yes. You are thinking in terms of the modern literary phase of math which includes writing. I can conceive of a pre-literary phase of math which is comprised only of verbal counting.

When Stonehenge was built, it must have been with verbal-only pre-literary math. Although to build the pyramids the Egyptians did have and use writing of some kind.

I agree with your definition though.

[mikema63]

Time certainly exists, it even bends.

Hong wu, when you start talking about two subatomic particles there is no imprecision. Electrons are perfectly identical to the point that the theory was put forward that there was only one electron in the whole universe and it was just going back and forth in time so that it was everywhere.

[Hong Wu]

Hong wu, when you start talking about two subatomic particles there is no imprecision. Electrons are perfectly identical to the point that the theory was put forward that there was only one electron in the whole universe and it was just going back and forth in time so that it was everywhere.

I think the metaphysical task in your life may be for you to some day accept how finite and ultimately weak this approach you are taking is. What are you going to do with that knowledge, blow things up? Someone already invented nuclear power, we just don't use it.

[Paradigm]

... Except that the "flow of time" does not exist either -- we just think and imagine that it does because the Earth rotates on its axis in the proximity of the Sun creating the illusion of what we call "days" and "nights" which seem to follow each other and thus somehow mark out what we call "time".

But time does not really exist. Same as math, which also does not really exist. They are both just all in our heads.

Fair hypothesis, but how is this supposed illusion produced? Day and night clearly exist, and to say that time is simply derived from them is begging the question. Why is it that we experience one after the other? Why do I not experience a flow from today to last night? If you are going to argue that time is an illusion, you're going to have to make a case for what produces the illusion. Day and night are not sufficient explanations.

[mikema63]

I think the metaphysical task in your life may be for you to some day accept how finite and ultimately weak this approach you are taking is. What are you going to do with that knowledge, blow things up? Someone already invented nuclear power, we just don't use it.

I think the biggest thing your going to have to accept in your life is that my humble finite and weak approach is the one that actually works and gets things done.