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khan

For those who wish they had a better grip on math (myself very much included) check out kahn academy.. it should blow your mind. Youtube the 60 minutes special on it.
 
I Think I'll Have Another Drink

I'm kind of a semi-geek math professor, well, Instructor until I get tenure, NEVER!

I used to be a fgure skater, so I'm only semi-geeky, but I make up for it by being completely in love with math. Not balance-the-checkbook math, but things like, "What IS a number, exactly? Were numbers invented, or did they always exist and were discovered?"

The Arabic Numeral "6" of course was invented, and denotes the concept of "sixness." We can count six sheep or six rocks, but does the concept "six" exist independently of the sheep or the rocks? Is that nerdy enough to argue about? So, did the concept of sixness always exist, and someone recognized "sixness" in six rocks and six sheep? Or did some nerd invent the concept of "sixness" and then apply it to rocks and sheep?" (Of course, the question about the concept "six" applies to all the other number-concepts.)

Anyway, there are at least two answers to the existence-of-the concept question, and mathematicians argure about them all the time. I think number concepts always existed and were discovered. That makes me the mathematical (but not literal) equivalent of a Creationist, or at least a proponent of Intelligent Design. Others think we made the whole thing up, and a few think we should call the whole thing off.

Anyway, I'm someone who know what she thinks, and I think I'll have another drink!
 
Mathematics is the sexual intercourse of the sciences. I pity the fool who reads those words and shakes their head in bewilderment.
 
tripleflip said:
... does the concept "six" exist independently of the sheep or the rocks?
Click to expand...

I see what you did there. Did six always exist? Did sex always exist?

Personally I'm interested to know if Beethoven's 5th Symphony will always exist. Let's imagine the sun goes super-nova and wipes out every trace of the Earth, every written piece of music, every CD and recording. Does the Symphony exist despite all that? You could ask the same things about six and sex for that matter.
 
What I like is that Bertrand Russell and Alfred North Whitehead wrote a book (Principia Mathematica) which took them up to page 379 to prove to their satisfaction that 1+1=2.

The best part about this is that Kurt Godel then managed to demonstrate that the proof was flawed.
 
tripleflip said:
I used to be a fgure skater, so I'm only semi-geeky, but I make up for it by being completely in love with math. Not balance-the-checkbook math, but things like, "What IS a number, exactly? Were numbers invented, or did they always exist and were discovered?"

The Arabic Numeral "6" of course was invented, and denotes the concept of "sixness." We can count six sheep or six rocks, but does the concept "six" exist independently of the sheep or the rocks? Is that nerdy enough to argue about? So, did the concept of sixness always exist, and someone recognized "sixness" in six rocks and six sheep? Or did some nerd invent the concept of "sixness" and then apply it to rocks and sheep?" (Of course, the question about the concept "six" applies to all the other number-concepts.)
Click to expand...

Doesn't it apply to all word-concepts, not just number-concepts? It seems to me that you're describing a question that's answered (in two different ways) in both Sam's first sermon in Lord of Light and the Appendix on Newspeak in 1984.
 
sun_sea_sky said:
What I like is that Bertrand Russell and Alfred North Whitehead wrote a book (Principia Mathematica) which took them up to page 379 to prove to their satisfaction that 1+1=2.

The best part about this is that Kurt Godel then managed to demonstrate that the proof was flawed.
Click to expand...

Well, actually, no. What Godel showed was that, among other things, that if a system such as Principia Mathematica (P.M.) is consistent, then there must exist undecidable propositions in it.

"1+1=2" isn't one of the undecidable propositions.

An example of an undecidable proposition may be Goldbach's theorem, which states that every even number is the sum of two primes. No even number has ever been found that is NOT the sum of two primes, but no one has ever PROVED LOGICALLY within P.M. that this HAS to be the case.

The significance of Godel is that he demonstrated that truth is a stronger concept than logic; that is, there may be true statements in a system like P.M. which cannot be reached by the logic of P.M.
 
tripleflip said:
Well, actually, no. What Godel showed was that, among other things, that if a system such as Principia Mathematica (P.M.) is consistent, then there must exist undecidable propositions in it.

"1+1=2" isn't one of the undecidable propositions.

An example of an undecidable proposition may be Goldbach's theorem, which states that every even number is the sum of two primes. No even number has ever been found that is NOT the sum of two primes, but no one has ever PROVED LOGICALLY within P.M. that this HAS to be the case.

The significance of Godel is that he demonstrated that truth is a stronger concept than logic; that is, there may be true statements in a system like P.M. which cannot be reached by the logic of P.M.
Click to expand...

I found one. 2 is an even number that is not the sum of two primes.
 
AdjectiveNounVerb said:
Doesn't it apply to all word-concepts, not just number-concepts? It seems to me that you're describing a question that's answered (in two different ways) in both Sam's first sermon in Lord of Light and the Appendix on Newspeak in 1984.
Click to expand...

Yes, of course.

My friends in the Modern Language Association have made careers demonstrating that the concepts expressed in language do not exist apart from the writer and reader. In other words, nothing exists but that thinking makes it so. The extreme of this view appears to be that monkeys with typwriters are excellent novelists, if only their READERS would bring more to the experience.

In mathematics, at least the way I use mathematics, the concepts are limited and recursive. For example, in the Peano axioms for positive integers, "one" is defined as the immediate successor of "zero." Then every other number is defined as the immediate successor of the previous number. However there are always undefined concepts because you can't define everything without being circular, as Plato will teach you. For Peano, the undefineds are "zero," "number" and "immediate successor." Most of us can grasp the essence of these undefineds. It must be admitted though that since math is built on undefined concepts, we mathematicians can't know what we are talking about.

For my friends in the MLA, the number of undefined concepts is very large, and limiting or bounding concepts such as "god" are not in favor. As a result, everything is what ever it you see it as being. They are probably right.

I suppose that's why, after a day of mathematics, I like to go skate. When I land a triple flip, the world comes back into focus.
 
AdjectiveNounVerb said:
Doesn't it apply to all word-concepts, not just number-concepts? It seems to me that you're describing a question that's answered (in two different ways) in both Sam's first sermon in Lord of Light and the Appendix on Newspeak in 1984.
Click to expand...

mynameisben said:
I found one. 2 is an even number that is not the sum of two primes.
Click to expand...

Oh, sorry, two IS a prime, by definition. (It is divisible only by itself and one.) Thanks for playing, though.
 
tripleflip said:
Well, actually, no. What Godel showed was that, among other things, that if a system such as Principia Mathematica (P.M.) is consistent, then there must exist undecidable propositions in it.

"1+1=2" isn't one of the undecidable propositions.

An example of an undecidable proposition may be Goldbach's theorem, which states that every even number is the sum of two primes. No even number has ever been found that is NOT the sum of two primes, but no one has ever PROVED LOGICALLY within P.M. that this HAS to be the case.

The significance of Godel is that he demonstrated that truth is a stronger concept than logic; that is, there may be true statements in a system like P.M. which cannot be reached by the logic of P.M.
Click to expand...

Which brings us right back around to what started this thread ...
attachment.php
 
tripleflip said:
"1+1=2" isn't one of the undecidable propositions.
Click to expand...

If I can just move on from "1+1=2" to "2+2=4", just to make things a bit more complex, I'll quote from Bertrand Russell here:

"But," you might say, "none of this shakes my belief that 2 and 2 are 4." You are quite right, except in marginal cases -- and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length is less than a meter. Two must be two of something, and the proposition "2 and 2 are 4" is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arise in which you are doubtful whether two of them are dogs. "Well, at any rate there are four animals," you may say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. "Well, then living organisms," you say. But there are things of which it is doubtful whether they are living organisms or not. You will be driven into saying: "Two entities and two entities are four entities." When you have told me what you mean by "entity," we will resume the argument.
Quoted in N Rose Mathematical Maxims and Minims (Raleigh N C 1988).
Click to expand...
 
Tripleflip, there's more to the MLA than just a style guide with what I always considered an unfocused and aesthetically displeasing citation format? Didn't know that.
 
How much is 1 and 1?

Let's address the "1+1=2" statement. The numeral one is a NAME for the successor of the undefined number zero, "s0." The numeral two is a NAME for the successor of 1, or the second successor of zero, "ss0." If you add the successor of zero to the successor of zero, you get the second successor of zero by definition: "s0+s0=ss0." If you substitute the names and suppress the undefined "0" you get 1+1=2. The other sums are proved by induction.

What Russell et al were concerned with in the cited chapter is CLASSIFICATION, which is the motivation for set theory. (Zermelo–Fraenkel et al)

"Two must be two of something, and the proposition "2 and 2 are 4" is useless unless it can be applied."

Note carefully that Russell does not claim that 2+2 is not 4. He points out that the proposition "2 and 2 are 4" is USELESS, not FALSE. In one of my earlier posts I pointed out that:

"It must be admitted though that since math is built on undefined concepts, we mathematicians can't know what we are talking about."

To make math work, you have to be able to count, and to count, you have to be able to classify:

"Is this thing one of the things we should count, or not."

If you are counting apples and ice cubes, you have to define a common set, which can be the set of "things" for example. Then the concepts of "ten-ness" and "eleven-ness" can be usefully applied to the set of things. Doesn't help much with the pancakes, purple aliens, or hats, though.

I admit that classification can be a very difficult problem, but it is a completely different problem from the abstract "oneness plus oneness is twoness," which expresses a logical consequence of axioms and definitions. Applying concepts like "oneness" and "twoness" to collections of objects (sets) is an important field of current mathematics.

Google "empty set" for some fun reading. Hint: It's better when stoned.
 
sun_sea_sky said:
I see what you did there. Did six always exist? Did sex always exist?

Personally I'm interested to know if Beethoven's 5th Symphony will always exist. Let's imagine the sun goes super-nova and wipes out every trace of the Earth, every written piece of music, every CD and recording. Does the Symphony exist despite all that? You could ask the same things about six and sex for that matter.
Click to expand...

Every event exists in time, as well as space. My friends in the physics department tell me that the simplest description of space time is Minkowski space. Every event is then a four-vector, <x, y, z, ict>.

To put this in ordinary terms, if I'm meeting some hot guy downtown, he has to tell me four things: the address of the building, for example 42nd street and 5th avenue, the floor, for example, 32nd, and the time, say, 3pm. Those are the four dimensions. Human beings are constructed to perceive three of the dimensions, the ordinary left-right, back-forth, up-down, as fixed and stationary. The fourth dimension is at "right angles" (orthogonal) to the others when you are at rest. Human beings seem to be able to percieve time as "flowing through" the three "ordinary" dimensions. This is not necessarily "how it really is," it's just how we perceive it.

An alternate, and just as justifiable, view, is that everything is absolutely still in four dimensions. You have to think about this a little to "get it." If the universe is a four-dimensional "blob," then time is just another dimension. It's nature's way of keeping everything from happening at once, from being an infinitely dense ball in the same time-space. But in fact, everything HAS happened, past, present, and future. It's all wrapped up in that 4D blob.

As humans, we can only experience existence as a 3D journy along some time line in the blob. We are fated to experience life as clock ticks, but that's not how it "really" is. Everything that has happened, and everything that will ever happen, is there in the "blob," always and forever, absolutely still, in some <x,y,z,ict> coordinate.

So Beethoven's 5th has aways existed and will always exist, along with the supernova and Earth and Taylor Swift. We can only EXPERIENCE what we call the "present" or our "absolute moment," but that doesn't mean the past and future don't exist and won't always exist.

This is not sci-fi, just a logical consequence of physics, and whether you choose to look at it this way or not. There are, of course, other ways to look at it. Modern physics has resorted to some very complicated math to try to preserve free will, among other things, but Occam's razor still cuts.
 
tripleflip said:
Every event exists in time, as well as space. My friends in the physics department tell me that the simplest description of space time is Minkowski space. Every event is then a four-vector, <x, y, z, ict>.

To put this in ordinary terms, if I'm meeting some hot guy downtown, he has to tell me four things: the address of the building, for example 42nd street and 5th avenue, the floor, for example, 32nd, and the time, say, 3pm. Those are the four dimensions. Human beings are constructed to perceive three of the dimensions, the ordinary left-right, back-forth, up-down, as fixed and stationary. The fourth dimension is at "right angles" (orthogonal) to the others when you are at rest. Human beings seem to be able to percieve time as "flowing through" the three "ordinary" dimensions. This is not necessarily "how it really is," it's just how we perceive it.

An alternate, and just as justifiable, view, is that everything is absolutely still in four dimensions. You have to think about this a little to "get it." If the universe is a four-dimensional "blob," then time is just another dimension. It's nature's way of keeping everything from happening at once, from being an infinitely dense ball in the same time-space. But in fact, everything HAS happened, past, present, and future. It's all wrapped up in that 4D blob.

As humans, we can only experience existence as a 3D journy along some time line in the blob. We are fated to experience life as clock ticks, but that's not how it "really" is. Everything that has happened, and everything that will ever happen, is there in the "blob," always and forever, absolutely still, in some <x,y,z,ict> coordinate.

So Beethoven's 5th has aways existed and will always exist, along with the supernova and Earth and Taylor Swift. We can only EXPERIENCE what we call the "present" or our "absolute moment," but that doesn't mean the past and future don't exist and won't always exist.

This is not sci-fi, just a logical consequence of physics, and whether you choose to look at it this way or not. There are, of course, other ways to look at it. Modern physics has resorted to some very complicated math to try to preserve free will, among other things, but Occam's razor still cuts.
Click to expand...

"he has to tell me four things: the address of the building, for example 42nd street and 5th avenue, the floor, for example, 32nd, and the time, say, 3pm. Those are the four dimensions."

I'm no math wiz and don't understand algebra but you've just asked forthree pieces of information:address, floor, and time.
 
tripleflip said:
Note carefully that Russell does not claim that 2+2 is not 4. He points out that the proposition "2 and 2 are 4" is USELESS, not FALSE.
Click to expand...

Russell was usually careful with his choice of words. For example in one case he was arguing against the existence of God, and suggested that perhaps the world was actually made by the Devil when God wasn't looking. Russell says, "There is a great deal to be said for that, and I am not concerned to refute it."

He didn't actually say he believed the Devil had made the world, just that he wasn't going to refute that belief.

So Beethoven's 5th has always existed and will always exist ...
Click to expand...

It existed before Beethoven wrote it? What if he decided not to? Or was he not able to make that decision?
 
sun_sea_sky said:
Russell was usually careful with his choice of words. For example in one case he was arguing against the existence of God, and suggested that perhaps the world was actually made by the Devil when God wasn't looking. Russell says, "There is a great deal to be said for that, and I am not concerned to refute it."

He didn't actually say he believed the Devil had made the world, just that he wasn't going to refute that belief.



It existed before Beethoven wrote it? What if he decided not to? Or was he not able to make that decision?
Click to expand...

Beethoven's 5th has always existed.

Free will is an illusion.

Think of any decision you made recently. Were you completely free to make it? If not, pick another one. Are you thinking about it? Good. Now, can you change it? No. It is "fixed" in time-space. At the end of your life, ALL your decisions will have been made. If you lived your life over, you would move through time-space along your time line from birth to death exactly the way you did the first time. There is actually no way to tell if you have been this way before.

Since everything exists in a four dimensional "blob" which is absolutely still, everything has always existed somewhere in the blob. We experience "free will" in our choices, but we are not free, we are bound in time. We are fated to experience our lives clock-tick by clock-tick as we move along our individual time-line which has always existed and will always exist.

"Free will" is an illusion which is necessary to our consciousness, which is also an illusion. Experiencing our full existence directly destroys the illusion, and our consciousness disappears, or becomes meaningless. Experiencing our full existence directly is equivalent to "seeing god" which is always fatal.
 
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