The Role of Mathematics in Human Life; What Is Math?

Great Debates
1

It can hardly be said without understatement: math is a part of almost every human activity directly or indirectly. It is apocryphally said that every human (or even dog) “does” calculus when it catches a ball thrown at it… Really, I think that is (possibly) overstated, but surely we do a lot of math and mathematical approximation in our everyday life. A short list:[ul][li]Estimating time to drive to various locations, near and (especially) far[]Balancing or roughly balancing the amount of money in our accounts[]Estimating the time it takes to pay off debts[]Deciding what size bookshelf, or how many bookshelves of a particular size, to purchase[]Counting caloriesDetermining what errors are acceptable[/ul]Really, the list seems endless. What strikes me about these things is not that they can be represented mathematically (as catching a ball) but that they are a direct and immediate application of mathematics and mathematical thought.[/li]
Let’s return to the ball-catching or ball-throwing example, because it is slightly more complicated than simple arithmetic and, I think, illustrates important points. There are three ways to look at the situation. One is that we have inductively “got a feel” for where a ball will land given its apparent speed and angle, and we can explain a ball’s motion given certain assumptions with math. Another is that we inductively “got a feel” for the math behind the motion itself, and have managed to formalize this motion with what is normally understood as math (that is, symbolic manipulation). Finally, the third possibility that I see is that we are actually doing the math (insert vague behaviorist “in some way” here as desired).

Mankind’s philosophical relationship with math has been very peculiar over the years. We’ve come, at various times, to try and discern what math means in a subjective and objective sense, and some have discerned a complicated game (formalists) while others a near-divine intuition (platonists). Still others have attempted to divine other isms to encapsulate the mathematical experience.

I think they have fallen short. I think the most suitable way to approach mathematics is as a language in a Wittgensteinian sense: an activity, a way-of-doing. This activity has no natural or artificial boundries. Any attempt to define mathematics qua mathematics or mathematics as an activity will result in the unfortunate circumstance of excluding obviously mathematical behavior. If we adopt Hilbert’s formalism, for example, balancing a checkbook and trusting that balance is a mystery. If we adopt mathematical platonism, we can be forced into a holier-than-thou stance where some of us have special access to an eternal realm, and balancing a checkbook properly is still a mystery.

Instead let us pursue mathematics as the grammar of certainty. Logic, set theory, number theory, algebra, the calculus, analytic geometry, ideal geometry: these, to us, are the language of certainty. To the extent that our activities or interests lie in certainty, our activities will tend towards mathematics. To the extent that I am certain of something, I can use mathematical speech or writing to show that certainty. A argument used as a tool for transferring certainty or knowledge has a symbolic form just as any word, sentence, or (public) thought. To the extent that my certainty can be shared, it is mathematical. (Contrast: “To the extent that I only feel certain, it cannot be shared.”)

I do mean to partially dismiss the symbology of mathematics inasmuch as English does not demand a symbology, but I do not mean to dismiss the symbology in that it is somehow distinct or seperate or useless in general. Where homophones fail, symbology succeeds; and where the mind fails to grasp large propositions holistically, symbology aids in that task. But it is important to remember that English is not as it is written, and math is not as it is done on paper.

We, as humans, do math all the time, but we have not all learned to speak math. Those who have not developed the ability to do math as an activity of symbolic manipulation are no less mathematical than the fact that I can’t speak French implies a failing on my part to speak a conversational language.

As with any language, the ability to master it (and I don’t mean “get a PhD”!) opens doors, enables pursuits, and increases one’s ability to assemble relationships and analogies. But also as with any language, its mastery comes from expediency or desire than a tabula rasa simply being written on.

Math per se is not a thing, nor is it strictly a tool for abstraction, a kind of representation. It is a language, and we learn it in the way we have found a need or desire to in our own daily pursuits. As a language, the meaning of math is in its use. For what does the symbol “this” mean? Well, we use it thusly. What does “x[sup]2[/sup] + y[sup]2[/sup] = 1” mean? Well, in these cases… and in these… and we treat it so.

The test for mastery of mathematics is the same as the test of mastery of a language: its use. We use this symbol as such, and those who use it otherwise are not violating reality but convention.

Mathematics, then, is not a part of, a result of, indicative of, or an ontology. The use of mathematics in science, for example, does nothing other than encode our certainty. Electrons aren’t points, even if our mathematical theories of electrons never ever say otherwise, anymore than I am literally solving a calculus problem when I catch a ball. We encapsulate the non-psychological rules of certainty with mathematics. We are sure electrons behave this way, and I don’t mean “we have a feeling of certainty”; I mean we explain electrons mathematically in order to demonstrate and transfer our certainty of how electrons behave.

Mathematics, then, is not a tool. It is not a metaphor, because if it were it would have to be “a metaphor for what it is” which is absurd (as if a sex scene was a metaphor for sex). The attempt to strain “tool” to include mathematics creates a container so big that everything is a tool. (And then what use is the word? What is is supposed to distinguish?) Math is not simply something we use to solve problems, because to say so we’d again return to the person throwing or catching a ball and suggesting, quite contrary to events, that the person is “really” doing calculus. The calculus equation in question requires the accelleration due to gravity, and then we’d be forced to suggest that the person “really” knows that value even if they cannot answer the question “what is accelleration due to gravity?” And that would abuse the verb “to know” which implies the ability to demonstrate (contrast: “to believe”, which makes no presumption of demonstration). So we can’t say math is a tool, or we’re still left holding the bag without explaining how people catch a ball.

No, the person catching a ball is “doing math” inasmuch as they have a behavioral certainty in their actions (again, not “a feeling of certainty”). “I knew the ball was going to be there.” Note that here the person demonstrates their knowledge by catching a ball enough times to satisfy the questioner–it is not required that he pull out a piece of paper. Note again the similarity to language, as we learn a language (say, from our parents), the test of our vocabulary is in a word’s use, not in our ability to scribble down a sentence and mark its position, its position’s name, et cetera.

As an activity, its test of compency or mastery is the performing of the activity itself in any way which that activity is used. Catching a ball, balancing a checkbook, estimating travel time: these are math just as much as proving the limit of an infinite series.

This is a testament to math’s generality. It is an abuse of the mathematical experience to suggest that people don’t know math because they can’t perform the symbolic manipulations, just as it is an abuse of the English experience to suggest that people don’t know English because they are illiterate.

The boundry between natural language and mathematics is also not distinct, which is further evidence of my suggestion. For consider, where do we place the following proposition, “Twelve times twelve is one hundred forty-four.” Is that a mathematical proposition, or an English proposition? Can we try and back away from the abiguity and suggest, “It is is a mathematical proposition in English”? But how did the certainty of math arrive in English if they do not have, as it were, a link? Is it merely a convenience that we have words in English for mathematical propositions? To me the answer is obviously no. Language as a behavior is not mere representation, or a tool, or an isomorphism of some kind. Mathematics is in various languages because it is a language and translation is possible, build on the bedrock of human activity.

Above, I noted, “Logic, set theory, number theory, algebra, the calculus, analytic geometry, ideal geometry: these, to us, are the language of certainty.” Now we see why the question, “Certain of what?” can lead us down strange ontological paths like formalism and platonism: the question is somewhat improper, and the answer is simply, “Certain of whatever we’re doing.” It also shows why math itself is so certain, because any attempt to determine its certainty is circular. Math is certain because certainty (as a public phenomenon) is characterized by math (as an activity).

To those of you who have a position on “what math is”, has my exposition made you rethink your stance? Do you agree with it? Disagree? Why?

2

Nope. Math is the same thing to me. Confusing and (thus) boring. 179-245 are the pages of a book to read, not whatever number it is. :wink:

On a more serious note, that was a rather epic post on maths. I would certainly define maths as a language in its own right. (Don’t ask me why I always write it maths, I’ve been around Brits too long) On the other hand, just like you don’t need to have a language to be able to think, I do not think that you need to understand math to, as you put it, “perform math” (in the case of the flying ball) - math is a language in that it allows expression of an idea (albeit, very specific ideas).

I would find a psychological study of this fascinating. A person who has no knowledge of maths or language being able to do basic calculations - I don’t think that you need to know 1+1 = 2 to know that 2 apples are better than one, for instance.

I don’t know what I’m getting at, and I’m nervous around numbers. Don’t poke me with a + sign please.

3

Now that is a *spectacular * OP… hurrah for math!

Continue…

4

erislover, have you read this?

5

I have not, II Gyan II. Sounds interesting.

6

Why was I so not surprised to see erislover’s name under this particular thread title?

Actually, I was thinking along these lines just the other day, but, alas, came to no interesting conclusions.

7

If you have II Gyan II, please share your opinion.

8

Well, I checked it out from my uni library alongwith other books, managed to read only the preface and half of chap 1 before I had to return it. I’ve reserved it again, so I’ll let you know at the end of next week. But from what I did read, they indicate that math isn’t transcendental or “objective”, just our conception of the world. But I’ll let you know after I’m done.

9

I disagree with this part. You say math is the language of certainty and symbols and their manipulation is encoding that certainty. Surely someone capable of working with certainty can master the representations that correlate to it. To take your English example, someone who’s illiterate, will most certainly, not know a great deal of vocabulary beyond the daily usage set of say, 1,000 words. They can’t open a dictionary and they can’t read philosophy (or Ulysses). What it comes down to is what does it mean to know something? Math and English are both vast and branched systems. A person doesn’t have to know abstract algebra to be said to “know math”. If you define what does knowing math mean, your statement can be better evaluated.

10

I agree with the OP - math is very language like and not some entity that exists beyond physical reality dictating it. I think that it doesn’t fit the mold of spoken languagess, but that isn’t what the OP requires of it.

My only issue with what was said is that you don’t do calculus to catch balls or move. The physical stuff that enacts the motions does a lot of input/output mapping that can be described by calculus but there is no mental calculus being done. Not even at a subconscious level. If you want to claim it is math then even nonsentient things do math in this way, and I don’t think that that was the intent of the OP.

11

A tad on the long winded side, and I’m not sure I even understand what the debate is, but the above statement I find confusing. Dogs are really good at catching balls. Are you suggesting that dogs are doing math when they catch a ball? If so, I would disagree.

12

The OP is a bit over my head, but I’ll take a shot at it.

I disagree with what I think is the thesis of the post: math is the grammar of certainty.

I always thought that in modern math, you start out with a set of axioms, and as long as you can’t derive a contradiction from them, it is as valid as Peano Arithmetic or Euclidian geometry.

Therefore, math doesn’t even have to be certain. It can totally contradict reality.

13

Well, I believe it is possible, yes, if they have some kind of reason to. But most manage fine without it.

What makes you say that? Do you think we only had around one thousand words before we developed writing? Do you think ~1000 words is sufficient for getting along? In The Language Instinct, Stephen Pinker suggests that the average high school student knows more words than was used in all of Shakespeare’s plays. Do you suppose this is only because they are literate? Certainly literacy increases a vocabulary… but is that to suggest that literacy of some words came before the words themselves? That would be strange.

For me, knowledge implies the ability to demonstrate whatever it is one knows in whatever way such things are (conventially) demonstrated.

But the point is that such a thing cannot strictly be said. We can set certain boundries, but they will not be definitive. We can suggest math is simply the formalistic activity the educational system brings us up to think, yet this defies all the practical application of math. Is only discrete mathematics real because we humans have finite limitations? But then what are we doing when we show that there are an infinite number of prime numbers? Knowing anything includes an ability to demonstrate one’s knowledge, through example or application, as the case may be. The engineer might build a model to show his grasp of the formulae, the grad student might elaborate on the consequences of a proof, the statistician must conduct proper polls and assert correlation at the proper times within certain bounds… yet these are all mathematical behaviors. Mathematics bleeds out from the core platonic ideal to the hopelessly mundane: figuring out if I brought enough cash to the grocery store so I don’t attempt to purchase more than I can afford.

Precisely! I did not mean to suggest otherwise. Calculus is an activity dealing with equations, and how the whole changes with respect to the components. It is traditionally done on pencil and paper. We can use calculus to describe ballistic paths… like a thrown ball about to be caught. It is a testament to our own certainty in such activites. Yet though I know where the ball will land so I may catch it, I am not doing calculus. But my certainty is no less real because of that. Calculus is an artifact of certainty, it is a particular kind of certainty. We use the calculus to assure ourselves or others of things. It would not be surprising, for example, to play a game of catch than have someone suggest, “You know that we can predict where the ball will land? See here, I have this equation which demonstrates its path…” But why isn’t it surprising that we can make such an equation? Because we are already certain of such things. That we may encode our certainty in such propositions is no more surprising than that the certainty was already there and demonstratable in another way. And since most humans are already convinced of the ballistics of thrown balls, there is no need for most of us to understand the other way we might suggest such a certainty (which is the activity of calculus).

But if we, as two students of math (hypothetically), are aware of this other certainty, can’t we use it to demonstrate the path of other objects, like missles which travel farther than we can play catch?

Well, spoken language doesn’t fit spoken language. A great part of communication is nonverbal. But mathematics is highly verbal and linguistic in nature. It can encode information, represent concepts, demonstrate relationships… it is very grammatical, with a rigid syntax (which must be the case if it is to be the language of certainty). At its broadest, math is how we explain our certainty, what is reasonable, what qualifies as such and why. This is not purely verbal, but it does certainly have verbal components. (Such as the English sentence I gave as an example, was it a mathematical proposition or an English one?)

I cannot strictly agree or disagree that dogs are doing math when they catch a ball. We do not usually suggest that animals have such a thing as certainty, or that they anticipate specific events, and so on. But I do not mean to say they do not, either. I am pretty uninterested in canine intelligence. While they might respond somewhat similarly to humans, the hallmark of math as a human activity (for me) is knowing where a ball is going to be. While the dog surely seems to know it, if dogs can be said to know anything, it is not open to questioning like a human so the sorts of claims we can make about it are far more speculative.

14

I don’t believe that’s the kind of certainty most would be familiar with. We can depend on the results of formal mathematics whether or not we find any particular application for them. The testament of math’s certainty is not its correlation to reality. That’s why I reject the notion that math is just something that represents: it doesn’t have to represent anything. Just like a perfectly grammatical English sentence doesn’t have to mean anything: “The ice was boiling hot.” There is no application for this sentence, and it contradicts reality, but is it not English because of that? We find a need, and address that need through the application of various behaviors. Once those behaviors are established, they can find themselves applied in other circumstances. Language is often used for reporting (states, events, feelings, cautions, etc) but we wouldn’t suggest that non-reporting statements aren’t a part of language (like a work of fiction, which may also totally contradict reality).

15

Well, what do you mean by “certain”? I thought you meant that if we prove something through mathematics, regardless of the initial assumptions/axioms, it corresponds with reality. This is simply not true.

I totally agree. I wish your other posts would have been as clear as that.

16

That’s what I tried to cover near the end of the OP. What we mean by certain is that it is mathematical. We are satisfied that the point is made because of limited induction (he’s performed the behavior ten or twenty times without my assistance), because of its structure (it follows a logical argument), because of its relationship to pencil and paper math (applying a formula in certain new circumstances). The hallmark of certainty is math: math, in a very broad sense, characterizes certainty. It is larger than pencil and paper tricks, even though we can (and often do) use pen and paper tricks to explain why we can be so certain. Check my response to The Tim above, I think I was very clear there.

Well, it is a big subject and I tried to cover several different views. Not all of the post would apply to every person, you know what I mean? :slight_smile:

17

agiantdwarf, let me add to a portion of my response to The Tim. I said, “It would not be surprising, for example, to play a game of catch than have someone suggest, “You know that we can predict where the ball will land? See here, I have this equation which demonstrates its path…” But why isn’t it surprising that we can make such an equation? Because we are already certain of such things. That we may encode our certainty in such propositions is no more surprising than that the certainty was already there and demonstratable in another way. And since most humans are already convinced of the ballistics of thrown balls, there is no need for most of us to understand the other way we might suggest such a certainty (which is the activity of calculus).” But you see, in this case math is the language of certainty. It is the form, the grammar of it. To express to you that I know how a ball will travel through the air, I may express it with pencil and paper mathematics, or I may demonstrate it behaviorally (i.e.-by catching the ball). I hope that made it more clear!

18

Of course, there is no way to determine the truthfulness of anything in mathematics, since you always have to assume *something. * However, defining math by circular logic (math -> anything certain, certainty -> anything mathematical) isn’t quite useful. It tells us nothing about the essential question in this thread: “What is math?”

Maybe I’m still misunderstanding you?

19

Certainty is characterized by math, the second part of your parenthetical. If we were to ask the question, “Is math certain?” then we’d run the circle. Which is why math is always certain. As my sig notes, “If a blind man were to ask me ‘Have you got two hands?’ I should not make sure by looking. If I were to have any doubt of it, then I don’t know why I should trust my eyes.” How do we get to be certain about math, say, pencil and paper math? I mean to say, how would I go about introducing doubt in your mind about the mathematical statement 1+1=2?

The grammar of certainty! :smiley:

20

[QUOTE=erislover]
A short list:[ul][li]Estimating time to drive to various locations, near and (especially) far[]…[]Determining what errors are acceptable[/ul]Really, the list seems endless. What strikes me about these things is not that they can be represented mathematically (as catching a ball) but that they are a direct and immediate application of mathematics and mathematical thought.[/li][/quote]
I think we need to distinguish between 1) Mathematics as consciously understood; 2) neurological functions that can be seen as a response to properties that are governed by the rules of number (e.g., seeing which object is farther without actually measuring); 3) and combinations of the first two.

Yes, we’ve got a feel. It’s neurological. So does a dog, as another poster pointed out. This is not related to a conscious understanding of mathematical principles.

No way. It’s pure neurology combined with trial and error that let’s us accomplish the task.

I find these three possibilities not to be distinguished very well; I really can’t see the difference between one and two.

You are describing the “problem” with either approach in emotional, pejorative terms and have not successfully argued against either.

This is not making sense. Mathematics is, at base, the incontrovertible rules of pattern, a subset of which is the rules of number. It is true that our understanding of these principles is certain, since they are provable. Yet we may be certain of other things that are not provable: I am certain just looked to the left now, but I can’t prove it, etc.

Backwards. If I use the rules of mathematics and I have done my calculations correctly, I can be certain that the result is true. Certainty is not something that is; it arises for a reason (memory, calculation, etc.). Also, “certainty” (a mental state) needs to be distinguished from “certainty” mean “true.”

This has quickly become a semantic hash. I don’t know what you mean by “share”–I can “share” my certainty about having looked to the left simply by telling you so. Perhaps you mean “compel” someone to share that mental state of certainty. We would then be getting into some heavy-duty epistomology…

Wrong. If someone tells me that pi = 3.15 even, that person is flat-out wrong.

No. Reality itself must obey the rules of pattern, which is what mathematical symbols encode.

No. We have a mathematical model for how electrons behave. We do not have, at first, a vague certainty about electrons that we must encode. Also, our mathematical model of how electrons behave bears no resemblance whatsoever to our neurological ability to catch a ball.

Neurological ability. If I put a 20 kg weight in your right hand and a 10 kg weight (that is the same size and shape) in your left hand, you and most people will be able to tell me which is heavier. Because all of Reality is based upon the principles of pattern and number, it’s no coincidence that your neurons are able to calculate correctly which is heavier (that is, it’s no coincidence that they are right). But there is no “understanding” of number implied in perceiving which is heavier, just as the sun does not need to calculate how much to gravitate the planets.

Bad example. Mathematical symbols are, as you said, a kind of language. Those who don’t know how to use those symbols may still be able to count verbally or on their fingers and use mathematical language to some extent. But if they know nothing, then, yes, indeed they don’t know mathematics. The neurological abilities remain, of course.

I’m afraid the exposition didn’t make much sense. I’m essentially a Platonist and have yet to understand what is wrong with my stance.