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The Role of Mathematics in Human Life; What Is Math?
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:
Let's return to the ballcatching or ballthrowing 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 neardivine 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 wayofdoing. 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 holierthanthou 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^{2} + y^{2} = 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 nonpsychological 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 questionerit 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 fortyfour." 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?
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125. 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. For why shouldn't I test my eyes by looking to find out whether I see my two hands? ~Ludwig Wittgenstein, On Certainty 
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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.
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What did the people in Sudan say about the Abu Ghraib scandal? "A few bad apples? You don't know how good that sounds right now! We're pretty hungry."  mnftiu Nothing says 'good luck' like handing off sovereignity and then running straight to the airport. Do we always treat sovereignity like it's a goddamn grenade?  mnftiu 
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Now that is a spectacular OP... hurrah for math!
Continue...
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"If you don't know an answer, a fact, a statistic, then ... make it up on the spot." Paul Watson, in Earthforce: An Earth Warrior's Guide to Strategy 
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I have not, II Gyan II. Sounds interesting.

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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. 
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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.
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"Quoting yourself is a sign of extraordinary arrogance."  The Tim My livejournal 
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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. 
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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? Quote:
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125. 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. For why shouldn't I test my eyes by looking to find out whether I see my two hands? ~Ludwig Wittgenstein, On Certainty 
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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!

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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? 
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Not in agreement....
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I think I see a way to make myself at least marginally more clear. For the example that keeps coming up, a thrown ball. I get the feeling that my objectors are picturing me suggesting that the person is actually "somehow" doing calculus or algebra in his head, when that implies a certain psysiological activity. I do not mean to imply that at all. In fact the last I heard, animals catch things like balls by following, apparently, a very simple routine that seeks to keep certain distances the same (like where the ball is in one's field of vision). That's all very interesting scientifically, but that is really not where I'm going. I'm saying, as a person, I know where the ball will land. One way to demonstrate that certainty is to catch a ball thrown at various angles and speeds to someone's satisfactionand what satisfies this other person is logical in nature. Another is to plot the course of the ball as with analytic geometry, work backwards from accelleration to a position plot with integral calculus, record a bunch of data and find a function that best fits it, etc. In either case, the person is satisfying mathematical interests, one on a broadly logical matter, another on a narrowly symbolic matter. Thermostats and dogs, as a rule, do not try to convince me of anything, as far as I can tell.

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I've admittedly only skimmed this thread, and I don't really intend to enter the debate, but I do have a couple of things that I thought could be added. In all honesty, I don't know that they're particularly relevant to the debate (in the sense that anyone's arguments will hinge one way or the other based on this), but I definitely think they are of general interest.
The first is the article Do Dogs Know Calculus?, by Timothy Pennings. To summarize, I'm sure anyone who has taken calculus at one time has seen the following problem: You're on one side of a river, and you need to get to a point on the other side (not neccesarily the point directly opposite). You can run on land at some given speed, and you can swim at some given (slower) speed. If you run along the bank of the river, then jump in and swim directly to the intended point, at what point should you jump in the river, in order to minimize the time taken? Well, this guy Pennings performed an (informal) experiment of throwing a ball into Lake Michigan, to have his dog fetch it. The point(s) at which the dog jumped in the lake were remarkably close to the "optimal" point! The other is something I read some time ago (sorry, no cite, this was 10 or 15 years ago, and I don't remember what the source was). The article stated that ball players (and presumably people and dogs in general), catch fly balls by chasing it in such a manner so that the ball (in flight) appears (to them) to travel in a straight line.
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...ebius sig. This is a moebius sig. This is a mo... (sig line courtesy of WallyM7) 
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erl: Are we setting roughly logical constraints on ourselves? "Hmm, every time I touch this, I get burned. He's like me; if he touched this, he'd get burned, too." Is this person developing a neural response, or is he doing logic, thinking logically [...]
Well, consciously verbalizing your reasoning as informally logical propositions isn't quite the same thing as unconsciously following neural signals that help you catch a ball. Sure, you could make the same sort of logical chain for ballcatching, along the lines of "Hmm, when it goes up higher it takes longer to come down, when the upward curve is steeper it doesn't go as far as when the curve is shallower", etc. etc. That still wouldn't be calculus, as you note, but it would indeed be conscious logical reasoning. But you don't need even that much conscious reasoning to successfully catch a ball, and neither does a dog. 
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And anyway. II Gyan II, I don't see whether it matters that hunter gatherers have a much smaller vocabulary than, say, a medieval knight or a twentieth century high school student. The point is that our investigations into life, and our ways of living, gave, imposed, or otherwise was correlated with our increase our vocabulary. Our investigation into farming, consciously directed or intrinsic ability based on contextual need, led us to words about agriculture. Our investigation into certainties led us to mathematics; i.e.mathematics is the language of certainty. But language is also an activity, and so doing math is the the use or transfer of such certainty.

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I think I can explain what erislover is trying to get at. Math isn't a magic sky pixie. Math is a mode of expression of facts of the universe but it isn't a required fact of the universe anymore than English is. There is no secret reality of computation where the math is being done to create what we see.
What this means is that math is a language. My minor quibble is that it isn't a language like English, and I mistakingly used spoken language when I suppose social language would be closer to what I meant and not as loaded with regards to the debate. Math is a language that describes principles we hold to be required principles that we are certain of. Social languages are those that describe our social reality. I think that it is entirely possible that eventually math could develop to the point where it could be a social language as well. People could think in an endless stream of math about everything happening during their day. Math is not like that yet and I don't think that math is wired up in the brain in the same way that social languages are. I think it is wired with similar tricks and exploitations of existing structures that language uses but the structures in question differ.
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"Quoting yourself is a sign of extraordinary arrogance."  The Tim My livejournal 
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The Tim, very well said. Quote:

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They know when they can demonstrate it to me in a way that either would let me know, or would satisfy me that they know. This is the point I am attempting to drive home, not some strictly analytical binaryvalued state of "knowing", but a social phenomenon involving the investigation, conscious or otherwise, of our knowledge and certainty. If that does not encapsulate the word "know" for you, perhaps you could do me the favor of telling me what word or concept it does match for you so that we may continue.
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125. 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. For why shouldn't I test my eyes by looking to find out whether I see my two hands? ~Ludwig Wittgenstein, On Certainty 
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To quote my abstract algebra teacher, "This isn't math, this is religion."
...seriously, now, I think math is a simplified way of looking at the world. From the most basic counting numbers (1 = "one of something") to calculus (the ball problem) to the really abstract linear analysis business (the rules of math itself). The simpler the thing we're analyzing, the simpler the math, and the more complex, the more equations. And besides, there's also statistics, probability, and fuzzy logic, all of which are branches of math that deal with uncertainty. A football player can throw a football without knowing that the spin stabilizes the fall flight; a physist can describe all the relevant equations, but not be able to throw worth beans. However, a pilot better know at least some aerodynamics if he wants to fly a plane, or he wouldn't know how to deal with all the variables involved. (Of course, nowadays computers are doing most of the numbercrunching for us, but I digress.) 
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About the literacy subthread that's appeared it is sort of missing a point about social languages  they exist in both formal and informal modes. Formal English is written English. Formal speaking sounds like it is written, or at least it does ideally. Illiterate individuals have only indirect access to formal language. They have a full grasp of the informal language and can describe the world adequately. To think otherwise is to be profoundly ignorant of linguistics. In many cultures with writing the formal language is written language. Writing allows for complicated sentences that would be nearly impossible to come up with on the fly. These are sentences that take full advantage of recursion because it is possible to look back over a written sentence or to read it slowly.
The formal language of languages without writing involves special vocabularly, structures and patterns that require specialized knowledge of the formal language. It is optimized to be done without notation and so doesn't produce longer utterances, merely denser ones that can only be decoded by those who have a grasp of the formal language. I'd recommend Doing Our Own Thing by John H. McWhorter for an interesting take on this issue. I'm sure somewhere in my post there is something that applies to the main point of the thread.
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"Quoting yourself is a sign of extraordinary arrogance."  The Tim My livejournal 
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I mean that they can describe everything that occurs to them in their daily lives, can refer to hypothetical situations and can come up with new expressions to describe new experiences. They will be no more at a loss for words than any other speaker of the language.
As for the scope of the world I would say that in most cases there is a strong correlation with those who have a wider range of intellectual experiences and those who have learned the formal level of their language. In this regard more esoteric experiences are likely to have more representation in the formal level than the informal level. It doesn't mean the informal language user cannot experience an equally wide intellectual world it just means they do not sound as educated when discussing them.
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"Quoting yourself is a sign of extraordinary arrogance."  The Tim My livejournal 
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Linguists interested in language use in a natural setting get people to agree to carry around tape recorders on their person, most of these studies seem to have been done in Britain which seems to have laxer rules about recording people who aren't entirely aware of it. Combine this with anthropological data on cultures without writing to get a full empirical look at it.
However it seems that this isn't all that is required to verify it. Formal language is developed by a culture, taught as a specific additional skill to the informal language and not mastered by all members. Not even literate individuals are masters of the formal language of their native language. It is rare to encounter individuals that cannot employ language for their daily needs and then some. People enjoy talking a great deal and about a great deal of issues even if they do not delight in the formal language. These facts imply that the formal language is not required for function. My question to you is what is a concept that a literate person can express that an illiterate person cannot? What concept requires the ability to produce long sentences with numerous nested clauses in order to be expressed? Think about having a philosophical discussion with your friends in person. If you wrote down a transcript of what was said would it look like a thread in Great Debates? 
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Does man select coincidental numbers to explain the Universe or do numbers dictate and reveal the absolute explanation of the mechanisms of the Universe to man?
Simple. Does one and one equal two? Of course not. First of all we must delimitate the existence of "twoness" in the Universe. And we can't. Oh well... 
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I believe I have answered the "what is it to know" question enough times already, actually.

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