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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 1:30:29 AM |
I believe mathematics itself exists as a pure abstract concept, like a language, so let's explore the "mathematics as language" concept for a moment and see how it might relate to "infinity":
Let's take our concept of language in general and see where it applies. Now there are a lot of mathematicians one might call "Platonists". They would insist all mathematics exists in some analogy of Plato's world of forms and only awaits "discovery." The mathematical formalists (most mathematicians) would insist as I do that mathematics is a language, merely a collection of symbols connected by rules of "grammar" to create mathematical expressions. They would further assert (as I no longer do) that it is just a game they play and any correspondence with what we call "reality" is only the most astonishing of coincidences. Then we have the constructivists, who claim that the only mathematics that exists is the mathematics we can define and express without fear of contradiction. Basically, they don't allow things like "infinity" to exist because of the mathematical problems it creates. The "problem" created by the constructivists is the renunciation of much of modern mathematics. There is a fear running though the mathematical community that the costructivists are "terrorists" who, if they had their way, would throw the baby out with the bathwater, just to make sure there are no holes in the tub.
Forgive me for the long post. This should be a discussion about the philosophies of math and fundamentally how you view the world regarding mathematics. (I suck at math but love it because I suck at it. But I don't fear it because I love the fact that I actually have to work at it. So if you're reading this and you think "OMG Math" don't worry. This is about philosophy and how you view the world regarding mathematics. All opinions are valid.)
I personally am a Platonist with a twisted view of the formalist. That is I believe that pure mathematics is the study of standardizing complex forms and all our physical world is explainable through these forms. Allow me to explain:
From my formalist perspective I believe we have to understand math as a language.
When we use the English language we are communicating ideas and concepts symbolically through words. When I say "go pick up the hammer" I am communicating the concept of picking up the hammer. Before the action is actualized the concept existed as a tool to achieve a certain purpose or goal. Mathematics is the ability to use such a concept to achieve the desired goal. But to understand math we must understand it as a language
It could be said that a language is a system of communication that communicates Ideas and concepts for a desired effect or purpose. As such to understand mathematics we must understand it in the same way we understand the parts of speech and syntax of a sentence. The difference is that math is extremely rigorous and almost immune to false logic.
It follows that the study of math (pure mathematics) is the study of a system that communicates ideas and concepts in a meaningful and purposeful way. The concepts and ideas it communicates are models built on axioms or postulates and theorems (derived from axioms which operate within the boundaries of said axioms). Any perceived limit of math is due to the system we have built to understand mathematics. Nevertheless, this physical universe is purely mathematical 
My platonist view involves the flying spaghetti monster......enough said.
Please feel free to discuss. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 3:34:37 AM | @NerdStatus
"Since I exist then what I consider real must exist. It implies that if I truly believe that a chocolate covered pixie goat is terrorizing me then it must be true. To add weight to my argument consider out of body experiences, schizophrenia, delusions, paranoia or any such phenomenon where a person is absolutely sure it is truth despite it being false. There is no way for the person to know it is false unless the person believes it is false." But the mere fact that something exists within my reality means that it is capable of being understood. The universe exists in my reality.
What I'm trying to say concerning my mathematical view point of reality is that the most obvious truth is that I exists. For my reality to exist I must conceptualize it. Every observation is related to a concept or idea and rendered down into the symbolic language we call English (or any other system of communication). Symbolic in that we use words to describe everything albeit abstractly.
Mathematics is the same way only "The concepts and ideas math as a language communicates are models built on axioms or postulates and theorems (derived from axioms which operate within the boundaries of said axioms). "
If something is capable of being understood and the universe exists in my reality. To understand it is to have it rendered down to a form of language. If we use math as a language and math describes all forms and the universe is a set of forms to be understood then the universe must be mathematical.
That is I believe that pure mathematics is the study of standardizing complex forms and all our physical world is explainable through these forms
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 5:18:44 AM | Math is a human (animal) construct. It comes from distinguishing and categorizing objects. Babies can subitize at a very young age and can add and subtract (small numbers) at a few months old.
I personally believe that certain areas of math belong in a separate box. Anything that uses infinity needs a separate box. Calculus allows the calculation of an instantaneous velocity, which is an absurd idea. Should we get rid of calculus? Of course not. It's ridiculously useful, but it needs to go into a separate box from mathematics that does not rely on infinity.
Infinity is a concept, not a number.
The worst abuse of this that I've seen is Cantor's transfinite numbers. The idea of different levels of infinity is crazy in my opinion and I don't think the way Cantor got there is correct. I take issue with 3 parts and all because of how infinity is used.
1. That a one to one correspondence implies infinite sets are equal.
2. How a one to one correspondence is defined.
3. His diagonal "proof".
BUT if there is a use for transfinite number (I haven't run across one yet...) then F it. Keep them. Use them. But put them in a separate box.
The important division is what is grounded in the real world and what is not. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 6:07:44 AM | Expression liberties generate information rivalry. The design which has to emerge in order to facilitate the reconciliation of information rivalry, is something I consider far more important than the information itself.
It is why I consider any language which seeks to remedy conflict with an authority on local levels, one that requires a ton of legwork in order to betray itself, as nothing more than a reprieve in learning the most inhibited of truths.
But then that is what humanity wants ... to delay the inevitable. To persist. 
What am I going on about now. I don't even... lol | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 6:59:34 AM | 1) What I realised, studying mathematics, was that all mathematical theorems are really just a conclusion of their proofs, because when their proofs work in a situation, then so does the theorem, and if in a situation that we thought the theorem was true, the proof turns out not to be, then neither is the theorem.
2) I also realised that if you read any proof, that it's just a shorthand way of expressing some piece of logic in English. Sure, it's written symbolically, but that's just the way it's written. You can make any mathematical argument in English, and about politics, and take any political argument, and write it out in mathematics, using mathematical symbols.
Put together, it means that ALL mathematical theorems are irrelevant. It is mathematical proofs that are relevant, and they are simply reasoning that we could find in any discussion, by written in shorthand, like Pitmans shorthand, just a different type of shorthand.
Mathematical theorems are only relevant, because they are the end result of that reasoning, that is a rule of thumb, for us to use easily. Really, there is only Pythagoras' Proof. We use Pythagoras' Theorem, because it's quicker to use a rule of thumb like a^2 + b^2 = c^2, than to re-prove the proof in any situation, at any time we need it. But, the Theorem is a shorthand for the Proof, because if we didn't have the theorem, and we did have the reasoning, then we'd reprove the reasoning (proof) in each situation, just like we do in any discussion.
Mathematics is a subject in which we recognise that we can generalise reasoning, and therefore just rely on some very easy rules of thumb to rely on. The difference between mathematics and other subjects, is that in other subjects, we don't formally recognise that we can make a rigorous generalised proof, with an attached rule-of-thumb, that makes our thinking much easier.
In this way, mathematics can be said to be a language. It is no different to writing your ideas out in Pitman's shorthand.
In this way, mathematics can be said to be a purely abstract form. We can argue about capitalism vs communism all day long. We'll never really prove which one is right. We'll each be able to dismiss the other's examples by pointing out that pure capitalism, and pure communism, are really abstract concepts. The same is for ideas. All ideas in their most general form, are abstractions. So, mathematics, being an idea that humans think about, is equally abstract. But, just like capitalism is practical in the form of the Law of Supply and Demand, it too is just as practical, with its laws, such as Pythagoras' theorem.
Really, you can debate about mathematics all day long, because you can debate about anything all day long, and produce the results you want. But, when it comes down to it, mathematical theorems are just shorthand for reasoning that we know works, and that are far more reliable than anything else we have in our society.
'Course, that p*sses people off, because that makes science not as important as mathematics. But that's life. You cannot make an omelette without breaking some eggs. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 11:06:37 AM |
all mathematical theorems are really just a conclusion of their proofs
You are describing a corollary; a theorem is a proved (or provable) proposition.
I also realised that if you read any proof, that it's just a shorthand way of expressing some piece of logic in English
Mathematics is a language in its own right. It can be translated into other languages (awkwardly in many cases), but it is my opinion that something is bound to be lost in the translation.
The difference between mathematics and other subjects, is that in other subjects, we don't formally recognise that we can make a rigorous generalised proof
Oh?...What about logic, or philosophy? Are these not formal systems relying on proof and rigor?
that makes science not as important as mathematics. But that's life.
Why is science not as "important"?...Is a scientist's work somehow less valuable because he may not be able to "prove" it? Is a scientist's work less applicable to reality because he chooses to explore reality rather than define it on the back of an envelope? The comment sticks in my craw, because it smacks of elitist snobbery, something I despise with a passion. There are mathematics, science, garbage and a whole lot of other things that people make careers out of working with. I for one will never assume any field of endeavor to be more or less "valuable" than any other. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 1:04:42 PM |
Mathematics is a language in its own right. It can be translated into other languages (awkwardly in many cases), but it is my opinion that something is bound to be lost in the translation.
Now I'll half introduce my flying spaghetti monster platonist philosophy.
If I am trapped in a room and the room has a door then its more likely that there is an outside to that room. It is less likely that there is no outside because the fact that there is a door implies that there is a reason for the door being there. The most obvious reason implies that there must be an outside to that room.
The logic is similar to mathematics and its meaning to the universe. In my thinking what we call math is not math. It is the translation of the nature of our reality into a symbolic form we can understand. Along the way things are lost or misunderstood or most often than not, too baffling to make sense.
It is my thinking that the rigorous language that is used as math is actually our best efforts in translating the "pure" language of this universe. The logic here is that math as a language is constantly evolving. This is an indication that the math we use as a language is evolving to certain form (I've always wondered if there is a limit to how much math we may know, discover, or prove). In the platonist way I believe that math as a language is evolving to a form that is the actual language of the universe. So when given a physical problem I am of the opinion that there is always a provable answer. I just have to discover it (its fun to think like that in problem solving).
BUT, when I say universe I mean the physical finite universe and its structures. Spirituality and conscious life is something that seems to be immune to mathematics. Sometimes it seems like the universe exists just so that a conscious observer may observe it.
that makes science not as important as mathematics.
Thats kind of contradictory.
Physics to me is the mathematics of the universe. It is mathematics just like non-euclidean geometry is mathematical but uses a different set of axioms. So really if physics is a science and physics is the math related to understanding the universe then "science is not as important as mathematics" is a contradictory statement since physics is a form of math and is a science.
Also mathematics is a science just like physics is a science:
From wiki
Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns,[2][3] formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.[4]
From merriam webster
1 : the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations
Definition of science from dictionary.com
a branch of knowledge or study dealing with a body of facts or truths systematically arranged and showing the operation of general laws: the mathematical sciences. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 1:36:12 PM | It is my thinking that the rigorous language that is used as math is actually our best efforts in translating the "pure" language of this universe
I prefer to think of mathematics being LINKED to physical reality by axioms that appear to correspond with our view of it. (For instance the axiom of equality tells us that x=x. It probably isn't "provable", but it is so obvious {our observations of the world around us have ALWAYS shown us that a thing will be the same as itself, so one could take the mathematical axiom of equality as a "property" of reality that the universe has been "speaking" to us since day one and therefore a point of correspondence between mathematics and observed physical reality.)}
However, like Euclid's parallel postulate, whose negation allows for non-Euclidean geometries, even negation of the little "obvious" axiom I mentioned could allow the creation of a consistent mathematical system that may (or may not) correspond with aspects of what we perceive to be "reality." I personally believe the language of mathematics to be much "richer" than the "language " of our perception of physical reality. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 5:04:59 PM | Math is a language.
My dad was a(no other words for it) a rocket scientist who supervised some folks who only were fluent in math-couldn't match socks...He'd say not a language-THE language.
The other side of that coin is music,as one is usually harder to grasp.
I perform music without thinking about it
The math person has to think about music,but math comes without thought.
I think one is as abnormal as the other,only absent from the DSM 4 'cause both types are functional. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 6:34:21 PM |
I prefer to think of mathematics being LINKED to physical reality by axioms that appear to correspond with our view of it. (For instance the axiom of equality tells us that x=x. It probably isn't "provable", but it is so obvious {our observations of the world around us have ALWAYS shown us that a thing will be the same as itself, so one could take the mathematical axiom of equality as a "property" of reality that the universe has been "speaking" to us since day one and therefore a point of correspondence between mathematics and observed physical reality.)}
However, like Euclid's parallel postulate, whose negation allows for non-Euclidean geometries, even negation of the little "obvious" axiom I mentioned could allow the creation of a consistent mathematical system that may (or may not) correspond with aspects of what we perceive to be "reality." I personally believe the language of mathematics to be much "richer" than the "language " of our perception of physical reality.
I agree that mathematics is linked to what we would consider physical reality. I disagree that the language is richer than our perception of physical reality. But at one point I would have agreed with you. Simply because mathematics is still unable to explain...consciousness. We perceive consciousness and I could argue that my existence is more true than the axiom of equality. But the language of math is still unable to explain it. Actually I'm not even sure if it even explains true curvature. To be honest there is a lot our language of math is unable to do. If it were richer than our perception of reality then it would explain the most self evident truth...self.
My dad was a(no other words for it) a rocket scientist who supervised some folks who only were fluent in math-couldn't match socks...He'd say not a language-THE language.
There is a subtlety between the the phrase math as a language and math THE language. The subtlety is that math THE language is still in development. We are still finding things out and still proving conjectures. If the language was complete we wouldn't be researching it. Instead the math we use now is the language of physical reality we are translating into a symbolic form.
I perform music without thinking about it
That's awesome! so do I.
When I perform music without thinking about it I hear the melody loud and clear in my head before I play it. I just loosen up my fingers and allow my body to play it. I perceive the melody before I play it. When the melody is played it can be quantified measured and communicated through the language of math unto a score. Is the melody "richer" with the score or is my perception of it (the melody I hear in my head) "richer"?
I would argue that my perception of it is much richer as it includes personal subtleties I cannot express on the sheet music. That is until the sheet music becomes able to express such subtleties. Till then its left up to interpretations. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 7:07:44 PM |
The other side of that coin is music,as one is usually harder to grasp.
Are you saying that people aren't typically good at both? Because I'm super awesome at both. And I can match socks. But I walk into walls. And eat paint. Yellow paint. It's okay because it wasn't snow.
Math is a language that describes concepts like any other language. The concept exists independent of the language. The concept is still created by people and doesn't exist outside of your awareness of it. Math requires units (and unit consistency, though the units can be abstract). Any division into units is arbitrary. You can create a world (universe, whatever; or even a view of our world, universe, whatever) without division. That just is. A single unit. There. Now there's no math. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 7:15:17 PM | @ Exo
mathematics is still unable to explain...consciousness. We perceive consciousness and I could argue that my existence is more true than the axiom of equality. But the language of math is still unable to explain it. Actually I'm not even sure if it even explains true curvature. To be honest there is a lot our language of math is unable to do. If it were richer than our perception of reality then it would explain the most self evident truth...self.
I think you ask too much of mathematics. It is only a language. By itself it explains nothing (it has no mind or mouth). It is the people working with the language who might use it to explain things. It may be that the words and rules to prove and explain consciousness exist already in many languages, but nobody has put them together yet in a convincing fashion.
As for your existence, you can't prove it (except perhaps to yourself), but it seems so obviously and intuitively true that you could probably consider it as axiomatic, even though you (presumably) don't know what it means to exist.
I'm not even sure if it even explains true curvature.
Define "true" curvature.
there is a lot our language of math is unable to do.
On one level, I would agree with you (there is no panacea), but on another, I would differ. Math in itself does nothing, but it IS a VERY rich toolkit, and the toolmaker has only the merest inkling of how to use them all. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 8:10:09 PM | This should be a discussion about the philosophies of math and fundamentally how you view the world regarding mathematics. (I suck at math but love it because I suck at it. But I don't fear it because I love the fact that I actually have to work at it. So if you're reading this and you think "OMG Math" don't worry. This is about philosophy and how you view the world regarding mathematics. All opinions are valid.)
I never, in a trillion years, would have guessed that you suck at math! Why I thought you were a math genius, exo! My preconceived notions of you have been crumbled...
The only reason I passed algebra in high school, was because I wrote a fifteen page essay on "Why I Hate Math", complete with illustrations. My instructor was so amused by my creative writing, that he counted it as a double grade, thus allowing me to pass by the skin of my teeth. Geometry, however, was easy...
I personally believe the language of mathematics to be much "richer" than the "language " of our perception of physical reality.
I actually disagree with this statement as well. If it were a richer language, it be able to quantify and qualify human emotions, such as greed, rage, love, friendship, etc...
One cannot explain these emotions, in algebraic form.
Math in itself does nothing, but it IS a VERY rich toolkit, and the toolmaker has only the merest inkling of how to use them all.
This...I totally agree with. It is a tool, just like spoken language. And I would go so far to say, that some toolmakers are more skilled at using their tools, than others. It is why I plague the more skilled with questions... 
Why is science not as "important"?...Is a scientist's work somehow less valuable because he may not be able to "prove" it? Is a scientist's work less applicable to reality because he chooses to explore reality rather than define it on the back of an envelope? The comment sticks in my craw, because it smacks of elitist snobbery, something I despise with a passion. There are mathematics, science, garbage and a whole lot of other things that people make careers out of working with. I for one will never assume any field of endeavor to be more or less "valuable" than any other.
Beautifully put...
Math is a human (animal) construct. It comes from distinguishing and categorizing objects.
I have actually been trying to make this point for about six months now, along with, philosophically, the proof of the existence of gravity.
And eat paint. Yellow paint. It's okay because it wasn't snow.
Okay. This was just funny...  | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 8:46:32 PM |
The only reason I passed algebra in high school, was because I wrote a fifteen page essay on "Why I Hate Math", complete with illustrations. My instructor was so amused by my creative writing, that he counted it as a double grade, thus allowing me to pass by the skin of my teeth. Geometry, however, was easy...
AHHHH! This kind of stuff drives me CRAZY. I hate the way math is taught (or rather, ISN'T taught) in school. We teach how to do math, not how to understand it. Algebra and Geometry are the SAME THING (or they are representing the the same concepts in different ways). Not on the surface, but underneath, to the point where they are interchangeable. Different things are easier in one or the other. (Have you seen how the Greeks used to solve quadratic equations with geometry? You don't [normally] learn that method in school...and the algebra for it simplifies to the quadratic formula, which is a generalized formula for completing the square...which is literally just that, geometrically completing a square...this feels like to much for parenthesis now...)
It's my belief that you hate math because it was taught to you incorrectly.
Math is actually a LOT of fun. For example, write out a set of numbers and try to come up with a function to generate those numbers from consecutive integers! Or look for new patterns! Or new ways to think about old concepts! Just don't make infinity a number...it's not! | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 9:41:28 PM | RE Msg: 8 by JustDukky:
all mathematical theorems are really just a conclusion of their proofs You are describing a corollary; a theorem is a proved (or provable) proposition.
I also realised that if you read any proof, that it's just a shorthand way of expressing some piece of logic in English Mathematics is a language in its own right. It can be translated into other languages (awkwardly in many cases), but it is my opinion that something is bound to be lost in the translation.
However, this isn't a formal proof. Neither you nor I can prove our views 100% today. We could argue this forever. I'd just rather say that in some ways, it resembles its own language, and in others, it resembles English.
The difference between mathematics and other subjects, is that in other subjects, we don't formally recognise that we can make a rigorous generalised proof Oh?...What about logic, or philosophy? Are these not formal systems relying on proof and rigor?
that makes science not as important as mathematics. But that's life. Why is science not as "important"?...Is a scientist's work somehow less valuable because he may not be able to "prove" it? Is a scientist's work less applicable to reality because he chooses to explore reality rather than define it on the back of an envelope?
The comment sticks in my craw, because it smacks of elitist snobbery, something I despise with a passion. There are mathematics, science, garbage and a whole lot of other things that people make careers out of working with. I for one will never assume any field of endeavor to be more or less "valuable" than any other. Of course it's elitist. But life isn't always equal in all ways to all people.
Say that you are in the middle of nowhere, and you need someone to drive you to the nearest town, and 2 2-seater cars show up, one with a professional driver, and one with a mechanic, and there is only room for you to sit in one of those cars. You have to pick: do you go with the driver, or the mechanic? If you go with the driver, then you'll get there a lot quicker. But, if the car breaks down, you're stuck. If you go with the mechanic, he only knows to fix the car if it goes wrong. But since he understands the mechanics of the car, he can figure out how the car works to drive it, not as well as the driver, and you'll take more time to get there. But, if the car goes wrong, he can fix it. So you stand a much better chance of getting to civilisation with the mechanic.
A driver still is important, just because mechanics exist. Each has his role to play. But you have to understand exactly what role you play, and how important your place is in the scheme of things. You won't always prioritise both. Sometimes, they'll argue over something, and you'll have to pick which to listen to. Sometimes, you'll pay one a lot more money than the other, and you'll get a better driver, and a worse mechanic. But, on the whole, you'll be a lot better off listening to the mechanic, and paying for a better mechanic than a better driver.
If the world loses all its scientists, we can recover. Mathematicians can still do the work of scientists, and figure out what they knew by experimentation. But, if we lose all the mathematicians, then scientists are unable to do their work. They can still do the experiments. But they're still reliant on mathematics to make them accurate, and without mathematicians to teach them the formulae they rely on, they won't have them to work with. So, you'll get a lot of experiments, but with very little in the way of conclusions, and what conclusions you will get, will be as reliable as saying that leeching cures all illnesses because it seems to make people better, when really it's just because giving blood gives you a sense of euphoria, a natural desire to drink plenty of fluids and get plenty of ailments, which any doctor will say is a very good idea when you're ill.
Life isn't completely equal. It's different. Scientists have their roles, and so do mathematicians. It used to work very well, because although you need to take mathematics more seriously, only 100 mathematicians would develop the maths needed for the work of 10,000 scientists. So you could afford to take them twice as seriously, and still 98% of your consideration and funding would be to science.
But, in today's time, that has changed. A lot of people see that as elitism, and feel that science has made more contributions than mathematics. As a result, much effort in maths has waned, and the consequences are that we are struggling to develop science. Science now needs lots of experiments just to prove one hypothesis, when one good experiment is necessary to prove the case, and mathematics includes using logic that helps you to pick the right experiment. Science often now has conflicting studies, when again, one good experiment would prove it conclusively. We keep being told how we are facing lots of difficult problems which we hope to solve in the future, but which seem to be taking an interminable time to solve. Again, one good change of approach using different mathematical tools, would solve the problem. But we're just pushing science and not maths. It's like having a car that only goes at 20mph, won't do a left-hand turn, and saying that what we really need is Michael Schumacher to compensate. We don't. We just one good mechanic.
RE Msg: 9 by exogenist:
that makes science not as important as mathematics. Thats kind of contradictory. Physics to me is the mathematics of the universe. It is mathematics just like non-euclidean geometry is mathematical but uses a different set of axioms. So really if physics is a science and physics is the math related to understanding the universe then "science is not as important as mathematics" is a contradictory statement since physics is a form of math and is a science. Also mathematics is a science just like physics is a science:
As a result, most people really don't even realise just how small physics would be without mathematics. As a result, you get plenty of kids who pay attention in science class, and very few who pay attention in maths class, and that's even with the fact that you cannot get into a university without a Maths GCSE for any degree in any branch of science, and that everyone taking any form of science has to study maths to a high level, even in Management Science degrees.
RE Msg: 10 by JustDukky:
I prefer to think of mathematics being LINKED to physical reality by axioms that appear to correspond with our view of it. (For instance the axiom of equality tells us that x=x. It probably isn't "provable", but it is so obvious {our observations of the world around us have ALWAYS shown us that a thing will be the same as itself, so one could take the mathematical axiom of equality as a "property" of reality that the universe has been "speaking" to us since day one and therefore a point of correspondence between mathematics and observed physical reality.)} "x=x" is provable. It needs to be, to prove that equality is an equivalence relation, and that is necessary to subdivide any group of things into their relevant values, and we know that we can do that intuitively. It's rather that the things that we see in the universe require certain mathematical truths to make any sense to us. However, sometimes, we are wrong, like with non-Euclidean geometries, and then we discover that the way we look at the universe is entirely skewed. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/1/2009 10:56:55 PM | Okay. I couldn't even write any poetry tonight, because all I kept thinking about was...math. Blast this thread! (Just kidding exo, you always come up with very thought provoking posts.)
So I reread the original post, to make sure I understood all of its fine nuances...
For me, that is how I can tell something is well written...when I reread it, I get something new out of it.
So I thank you...
this physical universe is purely mathematical
You are such a Pythagorean...
the flying spaghetti monster
A month, would not be a month, without the inclusion of the flying spaghetti monster. Speaking of which...have you seen her lately? She owes me three dollars for dry cleaning, after getting sauce all over my shirt the last time I saw her.
@Jeremy
I hate the way math is taught (or rather, ISN'T taught) in school. We teach how to do math, not how to understand it.
You are completely, and totally right. The successful understanding and application of a subject, depends entirely on the teacher's ability to teach...
I probably should have elaborated, that I took it again in college, and passed with a high B. (Which for me, and given my longtime mental block around math, is great.) The instructor, who was this tiny, older gentleman from Trinidad, did take the time to help me understand. He explained it by using an algorithm chart. Once the formulas were diagrammed, I grasped the underlying concepts and was able to apply them. He understood how I learned. It was like diagramming sentences, only using numbers...I got it.
It is just not as good of a story, as the one where I passed a math class by writing an essay on, "Why I Hate Math".
@scorpiomover
I will try to put this diplomatically...
Oh, screw it.
Your post made no sense to me. I read it three times. Your refutation is fallacious, and your analogies, well...let me provide one of my own. A scarecrow made of straw likes to argue a lot...get it?
I could pick apart your post, but I am tired, and I am going to go to bed...after I write some poetry.
However, I do want to address this:
Life isn't completely equal. It's different.
The second statement seems to contradict the first statement, since, what ruler are we using and what are we measuring exactly? Don't we all have (my first generalized broad sweeping statement...ever, I think) a different measurement of quality of life, therefore a different measuring device? How is one's contributions and worth, more important or, let us go ahead and say it, valuable than others? Your premise is based upon comparative value, as defined by the necessity of one's perceived usefulness, which might be needed at the time. (A little redundant wordplay, here.)
Anyway...I am not even sure I am making sense at this point...I am off too  | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/2/2009 12:46:11 AM |
I perform music without thinking about it.
Ditto. I was also fairly good at math when I was a kid. I made it up through calculus (that one gave me headaches), matrix theory but then I said "enough". I decided a piano keyboard was far more appropriate than (and I traded in) my trusty bamboo slide rule and never looked back. When mathmatics can describe the "mood" music creates, we'll talk.
You are completely, and totally right. The successful understanding and application of a subject, depends entirely on the teacher's ability to teach...
So true. Had I not come out of the womb loving music, the sweaty armpits of our boys glee club instructor would have surely turned me off the whole process. Successful teaching is a pure art form...learning to reach people on their level and not yours. But I digress.
He understood how I learned.
QED
more important or, let us go ahead and say it, valuable than others?
Which are, in and of themselves, purely arbitrary constructs.
Seems to me that philosophy and math are oxymorons. But I could be wrong. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/2/2009 3:50:43 AM |
For my reality to exist I must conceptualize it.
That's one point of view.
If we use math as a language and math describes all forms and the universe is a set of forms to be understood then the universe must be mathematical.
1)We use math to better understand our world, and make approximations.
2)Math doesn't describe all forms
3)We don't know that the universe is a set of forms to be understood
4)We don't know how mathematical (or not) the universe is, because we don't have a full understanding of the universe.
if any one measurement is infinte then all others are
I disagree. Some tings are finite.
Put together, it means that ALL mathematical theorems are irrelevant.
Tell that to NASA – I'm sure their mathematical theorems were totally irrelevant to sending out satellites, GPS, men to the moon, the computer you're typing on...
Mathematical theorems are only relevant...
Make up your mind. They're relevant, or they're not. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/2/2009 4:07:52 AM |
As for your existence, you can't prove it (except perhaps to yourself), but it seems so obviously and intuitively true that you could probably consider it as axiomatic, even though you (presumably) don't know what it means to exist.
Exactly. So what I'm arguing is that the philophosy of math should be understood from the first axiom that "I exist". Its more like a point of view kind of thing. We all view the universe differently and no two mathematical constructs are the same. What I mean is that if I stand at a certain angle and described the path of a falling body, it will be mathematically different from a different observer from a different angle or point of view. But that's easy. Einstein was a genius because that's basically what he did. Instead of viewing the universe through the same perspective (the ether theory), he went to the side and viewed the universe at an angle.
A great tutor once taught me from using the Cartesian plane that he and I can say the same thing from different perspectives. There may be differences in what is said but the concept remains the same. He then asked me what is the difference from thinking that a bubble is still in a cup of water and the universe is moving around it (that is in a Cartesian 3D plane the position of the bubble is always 0) compared to the traditional approach of the bubble moving along a line. He argued that that the two differing perspectives create equivalent but differing mathematical models. He then left me with the most annoying question of my life! What happens when you bring the two perspectives together?
I thought...hold on a sec, you can't! He then said think about it...
Its really hard to explain but I hope my argument is coherent. You asked me to define "true" curvature. If I defined true curvature I would use the mathematical language we have now to define it (lets eliminate the true from true curvature). I would probably start with the axiom two points create a straight line and probably end with the a curved path is a collection of intersecting straight lines. But is this the only way to define curvature? No, but quite simply it sort of works. I could use the conjecture a curved path is a collection of intersecting points to prove the existence of pi as well as a number of things. I'm sure I could even get to the calculus from this point. I could then tidy up the conjecture and say a curved path is only curved if the points at intersections is an infinite set. The nature of the curvature is defined by generalized rule of the set. Shoot I can do a lot more with that conjecture.
But the fact remains, is this the only way to define curvature? What if I viewed it at an angle like Einstein and brought it together with the traditional perspective? Maybe it would illuminate something we never knew before....maybe.
Then it comes to philosophies. Either you're this or that. But what if its everything and not just the Platonist, constructivist or formalist. What if they are all right but just from differing perspectives?
Diva in the post how do you know that you know said something profound. She said the best way to know is to collaborate. Being my usual self I interpreted in a twisted way. Maybe philosophies should collaborate as well. Quantum mechanics has been kinda trying to tell us that. At the most fundamental level don't expect the "either", or "or". Expect the "either", "or" plus "and" and everything all together; combining equivalent perspectives to continually illuminate things we never knew before. To me all philosophies is just a pinhole to view reality.
Math in itself does nothing, but it IS a VERY rich toolkit, and the toolmaker has only the merest inkling of how to use them all.
Hmmm...and again a different perspective. Let me collaborate and agree. Math is a tool kit and it is a language. If it is such then we are still left with defining that thing that uses the tool kit to create vast and beautiful mathematical structures that describe and communicate certain observations of our reality.
Now It makes me wonder. If math is a tool can it ever transcend the thing that uses it? If it can't, which seems to be more true than it can, then doesn't it mean that, that thing can never be quantified or measured or defined absolutely with math? If it cannot, which seems to be more true than it can, then isn't it more true to say that math is never more rich than the perspective of the thing using it. It is therefore rich in the sense that it achieves a great deal but not rich in the sense that it communicates the entirety of our reality. The thing that uses math however is capable of communicating much more than math is capable of communicating since it transcends math. What do we call the process this thing uses in perceiving this reality in a way that is more richer than the tool it uses called math?
Lets call this process belief.
I prefer to think of mathematics being LINKED to physical reality by axioms that appear to correspond with our view of it.
the important phrase here is "appear to correspond with our view of it". Then aren't the axioms which link physical reality to the appearance of our view of it little statements of beliefs of the thing that uses math as a tool? If they are, which appears to be true, then math truly is a tool.
Lets unite this view with mine:
I believe that pure mathematics is the study of standardizing complex forms and all our physical world is explainable through these forms.
Let a form be a collection of axioms that create a mathematical model/structure or the most relevant word describing the collection of theorems and operations derived from such axioms. To standardize a form is to communicate it through a language agreed upon by a set of people.
If math is a tool then pure mathematics is the study of the tool. It follows that applied math is the art or study of using the tool. But can I then not say that if axioms are a component of math and we get these axioms from our view of reality that the reality is communicating these axioms to us which we translate into the language of mathematics? Isn't this an indication that physical reality is using its own math where we are observing it to discover its axioms (synonymous with finding the syntax of certain languages) and theorems?
So the view that
In my thinking what we call math is not math. It is the translation of the nature of our reality into a symbolic form we can understand.
Still holds true as well as the view that math is a tool. A combination of the two views is that maybe math is not only the tool for us humans but also the tool for the universe. Its as if we are foreigners in a strange land translating the language of the land into our own so we can understand it.
{quote] I never, in a trillion years, would have guessed that you suck at math! Why I thought you were a math genius, exo! My preconceived notions of you have been crumbled...
I suffer from imposter syndrome...
It's my belief that you hate math because it was taught to you incorrectly.
So true. So very true. They just slap a theorem, operation or technique on you and say here use this!!! Then they expect you to use it to infer something when you don't even know why it does what it does!!! | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/2/2009 7:21:08 AM | RE Msg: 18 by mtnwldflower:
Life isn't completely equal. It's different. The second statement seems to contradict the first statement, since, what ruler are we using and what are we measuring exactly? Don't we all have (my first generalized broad sweeping statement...ever, I think) a different measurement of quality of life, therefore a different measuring device? How is one's contributions and worth, more important or, let us go ahead and say it, valuable than others?
1) Is a mathematician BETTER than a scientist? No. Why not? Because EVERYONE's worth is important. Is a redneck's worth less than a scientist? No. Is a religious minister's worth or contributions less than a scientist? No. FOR EXACTLY THE SAME REASON. EVERYONE'S WORTH AND CONTRIBUTIONS ARE IMPORTANT. Is that the way people are acting? NO.
2) In terms of how their contributions benefit society. Do a mathematicians' contributions benefit society less than a scientist's contributions? NO. Are they treated equally? NO. There are Nobel prizes for science, but not for mathematics. Can we argue that Nobel prizes are based on Alfred Nobel's will, and we shouldn't add to them? No, because a Nobel prize for economics was added. Does mathematics get the same level of funding as science? No. Is mathematics taught in school as if it was as important as science? No. Is science suffering from a serious lack of mathematical development? YES. Is our world suffering from a general ignorance of mathematics in the general public? YES.
Our world IS elitist. It makes scientists more important than redneck plumbers, religious ministers and mathematicians, when by any measurement, that should not be true.
There really isn't any particular need to make mathematics more important than science, because as a general rule, mathematicians don't tend to kick up a fuss. But there is a great need to ensure that mathematics isn't treated a lot less importantly than science, and that need is not currently being addressed. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/2/2009 7:39:38 AM |
Our world IS elitist. It makes scientists more important than redneck plumbers, religious ministers and mathematicians, when by any measurement, that should not be true.
It's up to us to change that. It isn't the world that's elitist, it's us! All we have to do is stop being elitist and the "world" will change.
there is a great need to ensure that mathematics isn't treated a lot less importantly than science, and that need is not currently being addressed.
It's a matter of need with respect to resource allocation. Mathematicians for instance might need a computer system that costs, say a few million, while physicists may need an accelerator costing several billion. Should we "equalize" funding and cut back on physics funding so the mathematicians can get enough pencils & paper to last them several million years? Which reminds me of an old joke about the physics dept at a large university:
The head of the physics department goes to the dean of the university with the annual budget for the physics dept. "We need a new particle accelerator which will cost $10million."
The dean squeals. "Whoa!! That's too much. Why can't you be more like the math department? They only want Paper, Pencils and wastebaskets! Better yet; why can't you be like the philosophy department?...It doesn't even want the wastebaskets!" | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/2/2009 9:21:03 AM | RE Msg: 23 by JustDukky:
Our world IS elitist. It makes scientists more important than redneck plumbers, religious ministers and mathematicians, when by any measurement, that should not be true. It's up to us to change that. It isn't the world that's elitist, it's us! All we have to do is stop being elitist and the "world" will change.
there is a great need to ensure that mathematics isn't treated a lot less importantly than science, and that need is not currently being addressed. It's a matter of need with respect to resource allocation. Mathematicians for instance might need a computer system that costs, say a few million, while physicists may need an accelerator costing several billion. Should we "equalize" funding and cut back on physics funding so the mathematicians can get enough pencils & paper to last them several million years?
CERN is spending billions on the LHC. But give a few mathematicians a grant to find a more efficient way to accomplish the same experiments, and he is very likely to, because that is just the sort of problems that mathematicians solve. They take problems that are difficult, and find a way to make them more efficient. So, if we'd done that, before the LHC was started, we might have paid a few mathematicians $50,000, and then built a slightly different contraption that would have cost millions, and not billions. If the mathematicians were exceptional, they might have worked out an ingenious way to test the same experiments using nothing more than a matchstick. Remember, Richard Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of the double-slit experiment.
That might seem unbelievable for mathematicians to achieve such efficiency. But I studied OR (Operational Research). Oil companies employ OR to save them millions, by working out the most efficient way to send their oil through their pipelines and lower their costs as a result. Large companies employ OR to work out the amount of stock to send to each store that will maximise profits. That's the power that maths has. It can take something so incredibly complicated, that no-one could work it out, not even a supercomputer, and find another way to do it easily.
So, the real question is: how much money would we have saved, if we HAD funded mathematicians to find a better way to test the experiments that the LHC can test? | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/3/2009 5:22:41 PM |
Diva in the post how do you know that you know said something profound. She said the best way to know is to collaborate.
That Diva was a wise, wise intelligent woman... | |
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