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Science, Math, and Philosophy Discussions regarding science, math, and/or philosophy.

View Poll Results: Is mathematics discovered or invented?
Discovered 40 48.78%
Invented 19 23.17%
Both 23 28.05%
Voters: 82. You may not vote on this poll

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Old 02-29-2008, 01:50 AM   #151
madnak
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Re: Mathematics: Invented or Discovered?

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Now add a billion tomatoes to five tomatoes. Count how many tomatoes you have. Repeat on the surface of the moon and Jupiter.
I lost count though.
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Old 02-29-2008, 03:32 AM   #152
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Re: Mathematics: Invented or Discovered?

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I lost count though.
59. you were at 59.
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Old 02-29-2008, 12:21 PM   #153
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Re: Mathematics: Invented or Discovered?

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Madnak,

Are you leveling in this thread? I know it's bad form to ask but I really need to know.
He doesn't know. It's pure reflex.
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Old 02-29-2008, 01:08 PM   #154
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Re: Mathematics: Invented or Discovered?

Dammit, it's a good method for truth seeking, even if true that it's a pure reflex.

The real question is are we on the level. I frequently exaggerate a point or play devil's advocate, and decide where I went wrong later.

Everyone needs to level with themselves from time to time. If we take our truths as self-evident, then we become ignorant. I appreciate it very much when someone points out my ignorance.

It's our pride that tells us that we're 100% on the level... that we have a good solid read on things.

It may be seen as playing games, or marksmanships. But, it's an important tool.

If madnak is ignorant on this (and me too), I'd want to see how, and I'd like to know why.
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Old 02-29-2008, 01:37 PM   #155
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Re: Mathematics: Invented or Discovered?

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You tell me. You're the one positing that nonphysical "forms" of some kind exist in a "somewhere" you can't describe. I agree it's not grammatically well-formed. We're 2500 years beyond Plato and his naive ideas.
I'm not actually positing any such thing. IMO numbers (and this is true more generally for other mathematical objects) are nothing more than places in a particular structure. This also partly explains how mathematics is epistemologically accessible; we can experience structures which approximate say, the natural numbers, and extrapolate from there. I don't know why you assume they're 'forms,' and I really don't know where you get the idea that everything that exists has to exist somewhere.

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I can as easily say that I believe "truth" exists, or that there exists a "negative" and an "affirmative." Of course I don't - I think Platonism is all nonsense. But you're the one picking and choosing. If you want to consider it real, you get to - but if it's inconvenient, then you can claim it's nonsensical. Can you provide a rational distinction?
Can I provide a rational distinction between why I believe numbers are real and 'negative' is not (I won't talk about 'truth,' because that's a pretty major topic on its own)? Sure: basically, if mathematical objects don't exist, pretty much all of science is literally meaningless or false. The same is clearly not true of 'negative,' which leads me to believe I don't have an ontological commitment to it like I do for numbers.

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And I challenge your assertion that you can conceive of them being independent objects. All Platonists of all kinds claim to have some special independent knowledge of these supernatural "objects" they see - but I don't think they do. Plato believed that all chairs were just expressions of an essential "chairness," and he believe that he was "in contact" with this "chairness" - I believe that Plato's concept of "chairness" was actually derived from the chairs he interacted with, and that if Plato had never interacted with a chair, he would have no conception of "chairness." This is more an epistemological question than a subjective question.
I wouldn't really characterise myself as a Platonist, to be honest. Have you ever read any Frege, out of interest? Your bit here is making the classic mistake he rails against, confusing how we come to know about something with the knowledge itself. Yeah, if Plato had never interacted with a chair he would have no conception of 'chairness,' but that's not a good enough reason to think it doesn't exist (I don't think it does, but for other reasons). It's like saying that if Columbus had never interacted with America, he would have no conception of it; it's true, but irrelevant.

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If we perceive it all within a linguistic framework, as description rather than metaphysical "truth," then there is no contradiction. "Unit" is a label that human beings have devised because it's useful, just as "4" is a label that human beings have devised because it's useful.
"4" is a label for a number, 4 is a number. But yeah, this doesn't help your position - you say that 2 or 4 are only meaningful when applied to units; fine, but I'd love to see you explain what a unit is without recourse to numbers, or mathematical objects. Just calling it a useful label doesn't get you out of this one.

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No, but I can say the same about language. But this is nowhere near analogous - I'm not talking about maths, I'm talking about mathematical realism. Mathematical realism hasn't been instrumental in the discovery of anything. And I do think it's very much the same as theism.
What do you mean 'Mathematical realism hasn't been instrumental in the discovery of anything'? Like I said, pretty much all of science when taken at face value implies that mathematical objects exist.

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I think my responses in this post and this post answer these questions.
They are the same post. You keep bringing up this picture analogy, saying that mathematics is like a man-made picture of something. Well, what's the picture of?

Also, I don't think you've answered the Dirac point at all: how is it possible that this mathematics which 'only exists in our minds' can somehow predict how nature is going to work without observing it? Since when has physics been contingent on what goes on in our heads?
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Old 02-29-2008, 01:47 PM   #156
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Re: Mathematics: Invented or Discovered?

No humans, no maths.
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Old 02-29-2008, 02:49 PM   #157
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Re: Mathematics: Invented or Discovered?

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No humans, no maths.
So 2+2 will stop equalling 4 when the race dies out?
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Old 02-29-2008, 04:02 PM   #158
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Re: Mathematics: Invented or Discovered?

Mathematical axioms and framework are human inventions.

From this framework we make discoveries. Some related to the framework itself (like the discovery of irrational numbers) and some of which reveal basic "truths" (which may be approximations to varying degrees of utility) to some aspect of the universe.

The answer is "both" imo and I can't see a case for any of the other two alternatives.
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Old 02-29-2008, 04:03 PM   #159
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Re: Mathematics: Invented or Discovered?

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So 2+2 will stop equalling 4 when the race dies out?
LOL..Interesting angle. (pardon the pun)
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Old 03-01-2008, 12:00 PM   #160
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Re: Mathematics: Invented or Discovered?

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Originally Posted by DrunkHamster View Post
So 2+2 will stop equalling 4 when the race dies out?
If the human race dies out, 2+2 won't exist. There would not be anyone to figure out what it equals.
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Old 03-01-2008, 03:52 PM   #161
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Re: Mathematics: Invented or Discovered?

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I'm not actually positing any such thing. IMO numbers (and this is true more generally for other mathematical objects) are nothing more than places in a particular structure. This also partly explains how mathematics is epistemologically accessible; we can experience structures which approximate say, the natural numbers, and extrapolate from there. I don't know why you assume they're 'forms,' and I really don't know where you get the idea that everything that exists has to exist somewhere.
And in what form does it exist if it doesn't exist somwhere?

Also, that structures only approximate the natural numbers indicate that these structures aren't the natural numbers. The structures are clearly separate from the natural numbers.

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Can I provide a rational distinction between why I believe numbers are real and 'negative' is not (I won't talk about 'truth,' because that's a pretty major topic on its own)? Sure: basically, if mathematical objects don't exist, pretty much all of science is literally meaningless or false. The same is clearly not true of 'negative,' which leads me to believe I don't have an ontological commitment to it like I do for numbers.
That's not true. We can apply arithmetic operators to "numbers" and get results that apply to science. We can also apply conditional and boolean operators to "affirmative" and "negative" and get results that apply to science. Of course, in science we must apply "affirmative" and "negative" to statements and propositions before we can manipulate them with operators - just as we have to apply "2" and "4" to objects or units before we can manipulate them with operators. But in both cases we can formulate these relations and manipulations without concrete referrents.

True AND false = false just as 2 + 2 = 4. True OR false = true just as 1 - 3 = -2.

And yes, most of science is false. We already know that. Everything Newton said has been proved false. And I suspect that relativity and quantum theory are false as well. I don't know whether it's possible to find truth, but it's more or less certain the science doesn't represent truth. This has been established in scientific philosophy since at least Popper, but probably long before then.

That scientific knowledge is false doesn't imply that it's meaningless. It's also closer to being true than anything else we have. It represents closer and closer approximations to the patterns that we observe, and in that sense is "accurate" even if not true.

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I wouldn't really characterise myself as a Platonist, to be honest. Have you ever read any Frege, out of interest? Your bit here is making the classic mistake he rails against, confusing how we come to know about something with the knowledge itself. Yeah, if Plato had never interacted with a chair he would have no conception of 'chairness,' but that's not a good enough reason to think it doesn't exist (I don't think it does, but for other reasons). It's like saying that if Columbus had never interacted with America, he would have no conception of it; it's true, but irrelevant.
I only know of Frege in that he tried to set up a comprehensive mathematics and failed due to Russell's paradox. But this again goes back to theism - just because I can refute all the arguments for God doesn't mean that God doesn't exist. And yeah, sure, "chairness" could exist. But if our conceptions of "chairness" are demonstrably contingent on our experience with chairs, then "chairness" is not necessary to describe our experiences with chairs (just as God isn't necessary to describe reality). More importantly, since different people have different mutually-exclusive conceptions of "chairness," some of our conceptions of "chairness" must fail to represent the actuality of "chairness" (just as the fact of mutually exclusive religious doctrines indicates that all religions can't be correct).

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"4" is a label for a number, 4 is a number. But yeah, this doesn't help your position - you say that 2 or 4 are only meaningful when applied to units; fine, but I'd love to see you explain what a unit is without recourse to numbers, or mathematical objects. Just calling it a useful label doesn't get you out of this one.
There is no such thing as "a number." 4 is a label regardless of whether we put quotes around it - it's an Arabic numeral, not a number. You're suggesting that it refers to something, but you can't show me what it refers to without invoking symbols and expressing it as a function of those symbols. You certainly can't show me any "quintessential essence" of 4.

I'd say a unit is a measured quantity used as the basis for other quantities. And ignoring the fact that my definition includes only words, no numbers (or even symbols for numbers, or even words for numbers as quanitities and numbers aren't quite the same), it still doesn't matter. Even if rulers exist (they probably don't), and even if the 12-inch mark on one ruler always matches the 12-inch mark on other rulers, that doesn't mean "12 inches" exists. In fact, relativity may indicate otherwise. "12 inches" is only relevant when compared to some other quantity - such as "24 inches." It has no independent existence. You can't measure out 12 inches in a vacuum, any more than you can measure out 4 of something in a vacuum.

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What do you mean 'Mathematical realism hasn't been instrumental in the discovery of anything'? Like I said, pretty much all of science when taken at face value implies that mathematical objects exist.
No, it doesn't. You'll have to support that claim if you want me to agree with it.

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They are the same post. You keep bringing up this picture analogy, saying that mathematics is like a man-made picture of something. Well, what's the picture of?
Oops. Meant to link this one, which is probably more relevant.

This picture is of a series of visual inputs that the painter interpreted as a mountain. The inputs are separate from the picture, and the mountain (if it can be said to "exist" at all) is separate from the inputs.

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Also, I don't think you've answered the Dirac point at all: how is it possible that this mathematics which 'only exists in our minds' can somehow predict how nature is going to work without observing it? Since when has physics been contingent on what goes on in our heads?
How can I predict that it's going to rain by looking at the clouds? My view of the clouds exists completely within my own mind - I am in no contact with the clouds. Certain information about the clouds is relayed to my eyes through photons/EM radiation, that information is translated and processed in my optic nerve, and then the view I have of the clouds is generated within my visual cortex (which is where it actually exists). My view of the clouds is clearly separate from the clouds themselves - the clouds could exist without my view of them existing, and my view of them could exist without the clouds existing. However, I know historically that when I see clouds, I tend to see (and feel and hear) rain shortly thereafter. Therefore it's useful for me, on the basis of this observed pattern, to respond to my view of the clouds based on the assumption that it precedes rain.

But the map analogy is the really appropriate one. If I draw a diagram of a chemical reaction and write "Krebs cycle" on it, that diagram isn't a chemical reaction. Nor is the term "Krebs cycle" a chemical reaction. Nor is the Krebs cycle itself a chemical reaction. The Krebs cycle doesn't exist - except in my mind. It's a label, a conception I've used to organize the interplay of molecules in order to make it easier for me to predict the results of molecular interactions. It's a feeble human attempt to describe the natural world. There is no "Krebs cycle" in nature - there are simply molecules. The molecules simply do what they do - there are no "cycles," no neat classifications. Nature doesn't need to divide glycolysis from phosphorylation from the Krebs cycle from everything else going on in the cell. In reality, it's just molecules. There no "this part of the cell does this, that part of the cell does that" - that's purely a human invention. There are no cells, certainly no "this part of a cell" or "that part of a cell," the only things that might actually exist are the molecules. But they don't exist either - they are another human invention, another pattern we've identified because we are limited and we must think of patterns as "real" even though they aren't - what really exists are just the fundamental particles, but oh, there again we have a human model existing in the human brain and not something that actually "exists."

We can find patterns in our sensory data and extrapolate those patterns. That is the only contact with "reality" we have. We humans have no more access to truth than dogs or insects. And the fact that we can identify, analyze, extrapolate, and communicate patterns doesn't give us a more fundamental understanding of the way things are than the dog's ability to track animals or the bee's ability to signal locations by dancing. Our ability to form abstractions is certainly useful, but it's hubris to think it means we've somehow touched the infinite.
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Old 03-02-2008, 01:07 AM   #162
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Re: Mathematics: Invented or Discovered?

I seriously get into a giggle fit every time I'm in math class! Every math problem we do, we just go back and forth... like making a slice from a cone. While going back and forth, if we find a contradiction, then we realize that the slice doesn't belong in the cone. Math is one really, really long frame of mind. But ideas like this are just childish and untrue.

Mathematics is beautiful to me.
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Old 03-02-2008, 04:11 AM   #163
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Re: Mathematics: Invented or Discovered?

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I seriously get into a giggle fit every time I'm in math class! Every math problem we do, we just go back and forth... like making a slice from a cone. While going back and forth, if we find a contradiction, then we realize that the slice doesn't belong in the cone. Math is one really, really long frame of mind. But ideas like this are just childish and untrue.

Mathematics is beautiful to me.
What class are you referring too.?? And maybe an example of a contradiction your speaking of. Are you dealing with cone calculus and parabolic equations.
Sounds like your making more of an error than finding any contradiction.
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Old 03-02-2008, 11:02 AM   #164
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Re: Mathematics: Invented or Discovered?

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What class are you referring too.?? And maybe an example of a contradiction your speaking of. Are you dealing with cone calculus and parabolic equations.
Sounds like your making more of an error than finding any contradiction.
Calculus, and not dealing with cones. I was trying to illustrate a point, but I will illustrate it another way.

We are traversing forward and backward between proofs. We go all the way down to limits, and all the way back up to differential equations. If we find a contradiction, then we realize the problem does not fit inside the cone of mathematics. Perhaps it's not the best analogy.
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Old 03-02-2008, 04:22 PM   #165
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Re: Mathematics: Invented or Discovered?

Very cool, sounds like you have a great teacher and glad to hear you enjoy this stuff so.
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Old 03-02-2008, 10:04 PM   #166
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Re: Mathematics: Invented or Discovered?

Thank you. I've realized forum hopping that there simply are no short-cuts to the truth. It's just one step at a time... from start to finish. wiki is a poor substitute for learning. I plan on getting my degree(s) in math, so I'm trying to learn how stuff gets proved from the top with ZF set theory and move from there. But, if I keep giggling in class, then everyone is going to think I'm completely bonkers (which I think is false). Do you teach math?
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Old 03-02-2008, 10:43 PM   #167
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Re: Mathematics: Invented or Discovered?

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Originally Posted by Flamingo View Post
I seriously get into a giggle fit every time I'm in math class! Every math problem we do, we just go back and forth... like making a slice from a cone. While going back and forth, if we find a contradiction, then we realize that the slice doesn't belong in the cone. Math is one really, really long frame of mind. But ideas like this are just childish and untrue.

Mathematics is beautiful to me.
donald duck in mathemagic land is the greatest film ever produced and i'll not sit here and listen to you slander its awesome power, good day sir.
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Old 03-02-2008, 11:48 PM   #168
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Re: Mathematics: Invented or Discovered?

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Thank you. I've realized forum hopping that there simply are no short-cuts to the truth. It's just one step at a time... from start to finish. wiki is a poor substitute for learning. I plan on getting my degree(s) in math, so I'm trying to learn how stuff gets proved from the top with ZF set theory and move from there. But, if I keep giggling in class, then everyone is going to think I'm completely bonkers (which I think is false). Do you teach math?
You know its funny, I kinda see there's two types of students, those who accept certain things. Like taking the derivatives of elementary functions ie: mutliply the variable by the exponent and subtract one from the exponent and go on from there, and then there's the few that really need to understand the whole proof of it in order to be comfortable with it.
Teaching is not my occupation but I'm a volunteer tutor. Very possibaly in the future though.
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