The names and notation we use for mathematics may be social constructs, but math itself is not.

No matter what you call it, 1 + 1 = 2. If you take a planet, and put another planet next to it, you have two planets. The circumference of a circle is pi * r * 2 no matter who is computing it, as long as they live under the same physics as we do. Even if we all believe it to be otherwise, it won't be that way.

Math is a social construct in the sense that if no-one is computing, math doesn't happen. You can say "2 and 2 is 4", but if you are not there, then there is no 'and', 'two' is meaningless or redundant, and the mathematical constants never appear except in their fuzziest form. Numbers are words and ideas, not physical laws.

Math is used to describe the physical universe, so it is shaped into a form that matches the universe as well as possible, giving it a framework. All things we would call math use this same framework.

But describing a Law and being a Law are different things. Math can describe a perfect circle, but that doesn't mean there is one. There is no circle that is as exact as the circle which is described by pi to 5 million digits. It is questionable if there is anything in the universe that really looks like 'two' -- especially if you were to describe '2' to 5 million digits.

Math measures things that exist, and it does this very well. But it can also do things that the universe that it is describing cannot. It is not confined by the same natural laws as the universe.

The truth in numbers is not fundamental. It is built by humans.

Magenta: The things we describe are truths and laws. There will be, anywhere and anytime, magnitude of change in a field as you move away from its source. Whatever describes this will be a mathematical idea, and we should be able to see a relationship between the described and the language describing it. But the phrase "gravity fields" is nonsense... there is only one, a dimpled one spreading across the universe. Breaking this thing down into tiny pieces is useful for us, but not something the universe does.

Likewise, digital math counts on being able to specify a certain amount; 1, or 2.1567, or Pi. But you could just as easily have about 1ish, maybe 2ish, a little over 3 -- it would often describe the real world better. But it doesn't move into the abstract well, and doesn't do human tasks well (e.g. money management). We start with an exact (impossibly exact) ideal and work from it. We have good reasons for doing this, but the fact is that mathematics is an over-engineered and idealized tool, not a map of a real territory... until we shave off some of the extra bits to make it look more map-like.

You have a mathematical truth as being an exact thing like 1+1=2, and you also have it as a real thing. They mix together, but I say they aren't necessarily bound to each other.

Ariels: They are True within the mathematic system. What does that mean? If physics describes reality, what could math be describing that is even more real? It sounds as though you are making The Truth a thing outside of reality -- e.g., a social construct.

Ariels 2:
"Show a bit of mathematical knowledge that is valid in one social setting and not valid in another social setting"

There is a Calvin & Hobbes cartoon in which Calvin is asked to answer 5+6. So he imagines a planet 5 big hitting a planet 6 big. Does he get 11? Nope. Nor I think, would an astrophysicist; but he is not in an 'astrophysicist' social setting, and his teacher is not pleased. But an even more clear example: dividing by zero is either something easy to do that we do all the time (there are 20 acorns on the ground; no one wants any of them; how many does each person pick up?), or absolutely forbidden.

The universe is a non-mathematical system. You cannot find a 'one' or a + or a factorial. Math doesn't exist in this universe. The words we use for math don't seem to refer to anything objectively real.

Try an analogy. A triangle is anything such that the sum of the angles in the plane is 2 right angles. A symbol is anything that can convey meaning to a sentient being. I draw a triangle. The sum of the angles in the plane is 2 right angles. I make a symbol. It conveys meaning to sentient beings. Therefore... Both math and symbols are true. But of course, neither math nor symbols are particularly true; they just sometimes correspond to truth, or close enough to build us a spaceship.

But numbers are physical laws. You can have one sheep, two atoms, three electrons... you can't simply say that an arbitrary quantity is a single quanta. An array of five by nine copies of TV Guide will always be 45 copies, no matter what number base or notational scheme you're using...

In 2001, Arthur Clarke pointed much of this out. The proportions of the monolith were a message. Although they weren't in human units of measure, the proportions were exactly 1:4:9 in the first three dimensions - the first three squares of integers. In fact, Dave Bowman found that it extended well past the first three in the higher dimensions.

Additionally, the ratio of the circumference to the diameter of a perfect circle will always be, in base 10, approximately 3.14159265358979323846264338, at least in cartesian spaces.

The Mayans used base 20. Porcupines would use octal. The Phoenecians used base 60. The Romans didn't even use a math system even closely resembling ours - their system of mathematics was a completely different social construct, and yet the mathematical truths were identical.

You cannot deny the fundamental truth in numbers.

(For the cluefully-impaired: I was just using TV Guide as a random unit. You could say the same about atom or quark. Volume is just an integration/approximation of the number of atoms in something. In fact, notice this: The derivative of the volume of a sphere with respect to its radius is its surface area. The derivative of the area of a circle is its circumference. The integral of a circle's area is the volume of a circular cone where the radius equals the height. And so on. If I'm not talking about the right sort of math, what sort of math are you talking about? I'm talking about what Saige was - that mathematics is a constant regardless of what number system or nomenclature you use.)

Mathematics describes truths. In this, it is almost unique. If I have a theorem, what it says is unconditionally true! Of course, we might be in error, and think that some proposition is proven, whereas in fact it isn't. But that's unavoidable in any human endeavour.

Even physics cannot lay claim to this position. Physics is (for the most part) a description of observed reality. As such, a physical law may be wrong, even though we have made no errors. Our measurements may not have been accurate enough, or we may not have measured the right thing, or any of the other problems which plague an empirical science may have cropped up.

Of course, mathematics is performed by humans, who are (for the most part) social beings. And the mathematical tools we develop influence the observations we can make of the real world. But our mathematical tools always have a privileged position: they are TRUE.

Tem42: "True within a system?" What "system" exactly is being used to describe mathematics?

And physics describing the world? Physics gives you (systematic) empirical knowledge of the world.

Physical knowledge is subject to empirical validation; when that fails, we have to change our physical knowledge. For instance, when Newtonian mechanics turn out to be insufficient to describe the world, we need to switch to relativistic physics and quantum mechanics.

But look at this amazing coincidence: All 3 theories are firmly based in mathematics! And NOTHING in the mathematics used to describe the newer theories which supercede Newtonian mechanics invalidates the mathematics used in the older theory. How come?

Quite simply, the mathematics is correct. Unconditionally. The sum of the angles of a triangle in the plane is 2 right angles. This is the sort of mathematical knowledge you need to describe classical physics. Note, however, that on a differential manifold the sum of angles of a triangle is not 2 right angles. For instance, on a sphere it is always strictly greater than that. Elaborations of this sort of knowledge are used in general relativity.

But here's the really cool bit: NOTHING in the theory of differential manifolds says that the sum of the angles of a triangle in the plane is not 2 right angles. How could it? It's TRUE. So one physical theory has replaced another (which is now considered "inaccurate"), but the mathematics is still perfectly good.

This also explains the mathematical character of physical law. Physics does all the dirty work of explaining empirical reality (that is, "what happens to be", as opposed to "what must be"). Naturally, what's left afterwards (to formulate the precise law) requires an absolutely true language -- mathematics.

Finally, a challenge to anyone who holds math to be a social construct: present an example. That is, show a bit of mathematical knowledge that is valid in one social setting and not valid in another social setting. That's all you have to do to convince me...

Muke: Your first example is very clear. Indeed, one cannot define division by zero consistently and retain the field structure of the real numbers. One can define division by zero by losing some of these properties (e.g. IEEE floating point numbers), but one is no longer working in the real numbers. What is the social construct here? The Heracles example, however, is simply incorrect. I do not understand how Heracles divides his 6 cows to nobody. There are certain properties that we would expect to hold when we say Heracles divides his cows. For instance, that each cow is given to someone (i.e. the division is in particular a partition of Heracles' cows). And if we accept these properties, then Heracles cannot divide his cows to nobody (since the empty set is not a partition of any non-empty set). This is one way of defining what we mean mathematically when we use the term "divide". You seem to hint at another possible mathematical interpretation of the word "divide". So you think there is more than one mathematical interpretation of "divide". Obviously you must agree on an interpretation. But this is merely because natural language is imprecise (among other things, it allows you to say things like "divide the cows to nobody" which are scarcely clear). If you look at something like the Monty Hall problem or the two envelope paradox, you get even better examples of imprecise or misleading phrasing in natural language. In all 3 cases, after removing ambiguity, nothing social remains. So if this were all, we could replace "math is a social construct" with "natural language is ambiguous", with which I would gladly agree.

Regarding Torddjen, that is a non-example. Unless I've done the math wrong, after going through the motions you describe, Torddjen now has 0 lasers (or "has no lasers", if you prefer to avoid zero too), and he owes !Kou 6 lasers and Ko'lei 2 lasers. This information is stored in MegaHAL's databanks, too.

The mathematician in Torddjen might want to know if he should get more lasers. After asking MegaHAL "how many lasers have I got, and how many do I owe", Torddjen sees that he has no lasers and owes 8. Since 0<8, Torddjen makes a note to head to the pawnshop and buy 8-0=8 lasers (working without 0, he'd simply skip the issue of the lasers he owns once he discovered he doesn't own any). Strapped for cash, Torddjen recalls an old gambling debt of 11 Adranian pints of Borgolias whiskey, drinks 5 (the recommended daily dose) and is left with 6 to pay at the pawn shop.

Yes, it would be more convenient to work with negative numbers. No, Torddjen would not have to change his business strategy, as he would owe the same number of lasers and have the same number of pints of Borgolian whiskey available to him. This is amazing! Even though MegaHAL has no negative numbers, its banking software gives exactly the same results as MicroHAL (which supports quaternions as its native format)! How come not using negative numbers changes nothing? Can it be that math exists in the real world regardless of the language we use to describe it?

Some examples of mathematical social constructs for ariels:

In our everyday math, division by zero is "undefined". But there's nothing to prevent someone from defining it, and if they try, their math will come out differently:

Heracles has six cows. He divides them among four children; they get each one cow, and two cows remain to him. He divides them among three children; they get two cows each, and none remain. He divides them among two children; they get three cows each, and none remain. He divides them to one child; that child gets six cows, and none remain. When he divides them to nobody, six cows remain to him.

Try and tell Heracles he can't keep his cows.

Similarly, we've invented something called a negative number. A society could easily create a mathematical system where such a concept is "undefined".

Torddjen has seven phasers. The system computer MegaHAL informs him that Ko'lei has a claim of two phasers on him, Abraxas four, and !Kou nine. Torddjen knows that Abraxas's cyborg thugs are very unforgiving, so Torddjen gives Abraxas his four phasers. Ko'lei is his friend and won't fault him if he delays another week, so he gives his remaining three phasers to !Kou. MegaHAL does not say that now Torddjen has "-8 phasers". That is meaningless. Every debt must have a debtor.

See if you can explain to MegaHAL who exactly Torddjen has -8 phasers in relation to, without getting both you and him thrown out the pod bay doors for doing math backwards.

I conclude that some math is a social construct.

"As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality." --Albert Einstein

ariels: The Torddjen example isn't mine, it's an adaptation of an example that came up in a linguistics mailing list I'm on in a discussion of number systems in our languages.

"Division" for Heracles means similar to what it does for us, i.e., the meting out of a quantity of objects into an equal number of groups. The difference between our "division" and Heracles's is the question of the remainder--for us, it is divided equally among the groups (5obj / 2grp = 2.5obj/grp) in which case dividing into zero groups is quite meaningless, but for Heracles it's simply not meted out at all (5obj / 2grp = 2obj/grp, R1) in which case dividing into zero groups is not meaningless. You are right that it is not the same as real numbers. The social construct here is that our society uses real numbers, and his does not (regardless of whatever consistency or truth-value there is in the systems themselves).

You are right that natural language is ambiguous. I think the ambiguity here is that I'm applying a different definition of math to the thesis of this node than you are. You might say I mean the science of mathematics is a social construct, and you mean that the mathematics themselves are not, and I would have no problem with that.

an update, after thinking about it more (leaving behind the original because people replied to it)

If you take a planet and put another planet next to it, you have two planets. This is only true because our language (a social construct and the basis for all social constructs) creates a category ("planets") that encompasses both of them.

You cannot count things until you categorize them. (or, you can't enumerate the contents of a set until you define the set.) And this categorization, even if it is so broad a category as "thing", requires a categorizer.

We start with categories. We begin to count. We begin to take generalizations about the real world and develop them as abstractions and axioms. We choose ideal, clean situations: we believe in the commutative property of addition, for example, because we have abstracted the idea of grouping identical, discrete objects that have no effect on each other.

Once we have developed our axioms, we construct our math on top of that. We don't choose the math (which is what saying "math is a social construct" sounds like) we simply discover the complex system that our simple foundational assumptions imply.

If I say math is a social construct, will I get lynched?

Mathematics are indeed a social construct. Remember, numbers, of themselves, have no physical existence. They only exist in our minds. We define one, for example, as the cardinality of any set which exhibits a bijective mapping to a certain set. There is no object or collection of objects which we can point at and say, "This is one". We merely create an idea and give it a name. We do this for all mathematical axioms and definitions. What follows, that is, the magnificent edifice of mathematics, is merely a restatement of these in a more elaborate form.

Math most certainly is a social construct.

The utilitarian look on math is that we use it to break down larger things into smaller, easier to grasp, pieces. This reflects our own nature as human being, as we look in on this world through a limited set of perceptions, viewing (or arguably projecting). More importantly, our perceptions, the concept of an object being separate from another object in one's mind, is needed only because of our inability to see everything, the universe and even more, as one single whole.

The example of putting two planets next to each other and insisting that a planet plus a planet must be two planets is flawed because of the word, "two." By specifying that anything has a number greater than one, you're identifying separate objects. The only reason we can distinguish between this and that is the advanced pattern matching software within our brain. This is explained in an excellent fashion by Alan Watts in The Book: On the Taboo Against Knowing Who You Are1; Watts shows a picture of the universe as it is, a seemingly arbitrarily chosen curvy shape, and then shows how humans see the world, the same curly shape with a grid over it.

One may argue that the number one, or two, or three, etc. in pure mathematics has no such connected value or associated material pattern, and is 100% abstract. Yet even then, the numbers are only a reflection of electrical impulses shifting within our brain, much as a computer shifts bits back and forth as it calculates. The only real number, the only true math, is the whole:


1In the Vintage Books Edition 1973 edition, this illustration can be seen on page 53.

This node chosen because it responds to above writeups.

Math is not a social construct and not a physical one. Math is a mental construct. There is no inherent relationship whatsoever between math and physics. Saige (and Greg Bear) are incorrect to assume that the value of Pi depends in any way on our physical surroundings. It does not; it is a property of certain idealized entities that exist solely in our minds.

Strictly speaking, any correspondence between mathematical constructs and reality is pure coincidence. Of course, in reality this is a bit different; people prefer to have applications for their thoughts. Additionally, our mind is of course strongly influenced by our physical and social environment. But there are branches of pure mathematics where corrspondences to reality are few and far between, if they exist at all (sometimes, they pop up quite unexpectedly).

The solution to the Torddjen example is that accounting is a social construct; the underlying mathematics is not. Similarly, the fact that the 1:1.618 (or 0.618:1) ratio is the basis of Greek architecture such as the Parthenon is social, the Fibonacci sequence is not, the mathematical rules which Fibonacci used are not.

Now that I think about, maybe Fibonacci was on the borderline. The application of the Golden Mean when depicting perspective in art is likely subjective to humans, but a CS professor once told me that perspective is learned; someone who grew up in a dense jungle where the visibility was limited to a few metres would have no sense of visual perspective. To what extent is our art and culture derived from biological traits common to all humans? Or to what extent is art about triggering pre-existing, default patterns in the brain, and to what extent is it about appealing to patterns developed in the mind through living within a society?

In summary, math itself is not a social construct, but applications of it may be.

There has been an interesting step forward on this subject in 1999: Jean-Louis Krivine, a french researcher in mathematical theories concerning AI, managed to prove that both mathematics and natural languages are subsets of Lambda calculus. It means that the human brain is an aggregate of many "neuron circuits" which each perform a basic Lambda Calculus function. This brings a quite shocking consequence: that what we call logic is a human construct. But in a sense, it is independent from the human mind, since such construct spontaneously emerges from the interconnection of neurons. The brain appears as a machine programmed in Lambda-Calculus, but it's physical laws that do the programming.

By connecting the two we get the idea that logic is a consequence of physical laws.

This result goes against the theories of dialectics that suppose that the concepts lying in mathematics and logic are independent from matter, and would exist (or still be true) even if the Universe did not exist.

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