The Opposite of Nothing Is/Isn't Everything

[Roger Bissell]

I don't know what silly things some Objectivists may have been expounding in mathematics, but I didn't read anything in ITOE that contradicted what I learned in math. I don't know where Roger gets this stuff about numbers being operators! "Plus" is an operator. A number is an operand.

Laure, please go back and read what I wrote, when I cited the Wikipedia article. I am using the term "operator" in regard to numbers analogously to the way it is used in regard to functions. We are just arguing terminology here. My point is that, whatever you want to call it, 0 does not do anything to other numbers. Not: it does nothing to them, but: it doesn't do anything to them. Whether you want to call that not being an operator, or something else, is just semantics. Please let's stick to the point.

Example in pseudocode:

Plus(a,b )

{

return a+b

}

Plus3(a)

{

return a+3

}

In the second one, 3 is a constant, or you can think of the function as being y = x + 3. The operator is "plus" and it is acting on the operands x and 3.

[Roger, you seem to be saying that 0 + x does not equal x + 0. If it doesn't, then what IS x + 0? If you're chugging along trying to solve an equation, and you get it down to:

x = 12 + 14 - 14 ...

x = 12 + 0 ...

Is your next step "Ah Ah Ah, x is undefined because you cannot add zero to anything"?

Or do you say x = 12?

You say the ~sum~ is undefined. But you still say x = 12. You just don't proceed beyond the 12. It is as if ~nothing~ followed the 12. Or, I should say, it is because there ~isn't anything~ that follows the 12.

The example you gave would actually proceed, from left to right (foregoing the obvious arithmetic shortcut of reducing 14-14 to 0), as: 12 + 14 = 26 - 14 = 12. The virtue of combining 14-14 is that you actually, by visual inspection, are able to reduce the number of calculations/operations from two to one. Now you realize, by inspection, that you can disregard the +14 and -14, because their sum produces a non-operator, 0, and you can stop with the 12.

I am saying that the sum of 0 and x is = to the non-sum of x and 0.

ATTENTION, ALL: this does NOT mean that I am endorsing additive commutativity of 0. The left hand of the equation represents a sum, but the right hand of the equation represents a non-sum. They just happen to be equal, because when you don't have anything and ~actively increase~ what you have by something, that gives the same result as when you have something and you ~don't actively increase~ it by anything. One is a sum, the other is not. That is why there is no such thing as additive commutativity of 0, even though the expressions 0 + x and x + 0 are equal.

REB

[merjet]

So, Dragonfly, in 2 x (times) 3, the operand is 2, and the operator is "multiply by 3"?? How do you square this with the terminology we all learned in grade school, that the multiplicand is 2, and the multiplier is 3? Or are you saying that all of our textbooks and teachers were wrong, and that the multiplier was actually "multiply by 3"? You're not making a lot of sense here -- or perhaps it is higher math that is not consistent with arithmetic terminology.....

I will chime in with Laurie here. 2 and 3 are both operands. x (times) is the operator. This is consistent with the following definitions:

http://www.yourdictionary.com/telecom/operator

http://www.yourdictionary.com/operand

You shouldn't rely on everything they taught you in grade school. I'd guess you were also taught you could not subtract a larger number from a smaller one.

OK, I did a Google search, and the second hit was the Wikipedia entry on "operator," and it defines it as "a function which operates on or modies another function." I am using it in an analogous sense, as "a number which operates on or modifies another number."

Your analogy derailed. Your first used "number" as an operator, and a number is not an operator.

I can't explain why the Wikipedia article doesn't address arithmetic signs more concretely, but some operators do operate on functions, such as the ones in calculus the article gives. It only addresses addition abstractly:

One says "addition operator" when focusing on the process of addition, or from the more abstract viewpoint, the function +: S×S → S. (Wikipedia)

[Laure]

...

That's ~precisely~ how I'd set it up. That is how we set up ~division~ (i.e., by bifurcating the cases), so why not make it consistent across the four arithmetical operations, to reflect the fact that 0 is not an operator for any of the operations?

quotient = dividend/divisor (if divisor not = 0)

else, quotient is undefined

(There ~is no~ quotient, because there is no operation. You can't divide SOMETHING by NOTHING.)

product = multiplicand*multiplier (if multiplier not = 0)

else, product is undefined

(There ~is no~ product, because there is no operation. You can't multiply SOMETHING by NOTHING.)

difference = minuend - subtrahend (if subtrahend not = 0)

else, difference is undefined

(There ~is no~ difference, because there is no operation. You can't subtract NOTHING from SOMETHING.)

sum = augend + addend (if addend not = 0)

else, sum is undefined

(There ~is no~ sum, because there is no operation. You can't add NOTHING to SOMETHING.)

I'm still waiting for someone to show me the flaw in this. Most of you are so convinced I'm an ignoramus. Please show me something I don't know that refutes the above, or that renders it fatal to the existence of mathematics.

REB

Roger, come on! X*0 IS defined. It's 0. X-0 IS defined. It's X. X+0 IS defined. It's X. As I pointed out in my other post, if you say these things are undefined, you are going to be stopped dead in your tracks when you get:

X = 12 + 14 - 14...

X = 12 + 0...

*freak out!! -- gaah!! -- undefined!!*

Dividing by zero is fundamentally different from adding, subtracting, multiplying by zero. Draw a graph of y = 1/x, for goodness' sake. It goes to infinity at x = 0. Now draw a graph of y = 1 + x. It goes to 1 at x = 0.

Didn't Ayn Rand say "A is A is all there is to metaphysics. All the rest is epistemology."?

I want to reiterate that numbers are not operators, they are operands. They're "nouns" if you like, while the operators are "verbs" (adding, subtracting, squaring...).

You compare adding zero to dividing by zero, when really what it's like is dividing by 1. What does it mean to "divide a number by 1"? You're NOT dividing it at all, right? But that doesn't mean that we say that x/1 is "undefined" - it's not, it equals x. Just like x + 0 is not "undefined", it's x.

Think "Occam's Razor", keep it simple.

[Michael Stuart Kelly]

(I have got to jump in here, even though Merlin and Dragonfly are handling things ably themselves, and even though I vowed I wouldn't post here anymore because I really don't like being accused of "not thinking with my own mind.")

Laure,

I don't recall ever doing that with you in mind since I never thought that of you, not even in our most explicit disagreements. If you interpreted my words to mean you, then a mistake was made, either with my rhetoric or your interpretation. (I suspect both.)

With respect to the part that pertains to me, sorry for the mix-up.

Michael

[BaalChatzaf]

ATTENTION, ALL: this does NOT mean that I am endorsing additive commutativity of 0. The left hand of the equation represents a sum, but the right hand of the equation represents a non-sum. They just happen to be equal, because when you don't have anything and ~actively increase~ what you have by something, that gives the same result as when you have something and you ~don't actively increase~ it by anything. One is a sum, the other is not. That is why there is no such thing as additive commutativity of 0, even though the expressions 0 + x and x + 0 are equal.

REB

Roger. Let S be the set of (integers, rational numbers, real numbers, complex numbers). The binary operation addition (+) is a mapping from S x S into S. It happens to be commutative and associative. That is:

1. a + b = b + a

2. a + (b + c) = (a + b) + c

The statement 0 is commutative is meaningless.

0 is a number. It is a special number. It is the identity for the commutative group (S, +), where S is defined as before.

Ba'al Chatzaf

Edited May 12, 2009 by BaalChatzaf

[Roger Bissell]

...

That's ~precisely~ how I'd set it up. That is how we set up ~division~ (i.e., by bifurcating the cases), so why not make it consistent across the four arithmetical operations, to reflect the fact that 0 is not an operator for any of the operations?

quotient = dividend/divisor (if divisor not = 0)

else, quotient is undefined

(There ~is no~ quotient, because there is no operation. You can't divide SOMETHING by NOTHING.)

product = multiplicand*multiplier (if multiplier not = 0)

else, product is undefined

(There ~is no~ product, because there is no operation. You can't multiply SOMETHING by NOTHING.)

difference = minuend - subtrahend (if subtrahend not = 0)

else, difference is undefined

(There ~is no~ difference, because there is no operation. You can't subtract NOTHING from SOMETHING.)

sum = augend + addend (if addend not = 0)

else, sum is undefined

(There ~is no~ sum, because there is no operation. You can't add NOTHING to SOMETHING.)

I'm still waiting for someone to show me the flaw in this. Most of you are so convinced I'm an ignoramus. Please show me something I don't know that refutes the above, or that renders it fatal to the existence of mathematics.

REB

Roger, come on! X*0 IS defined. It's 0. X-0 IS defined. It's X. X+0 IS defined. It's X. As I pointed out in my other post, if you say these things are undefined, you are going to be stopped dead in your tracks when you get:

X = 12 + 14 - 14...

X = 12 + 0...

*freak out!! -- gaah!! -- undefined!!*

[Clarification on 5/5/13: Laure, I say specifically that the SUM, PRODUCT, etc. is undefined. When the operations don't take place, there is still a COUNT of the items presented, so there is still a RESULT. That is why 1 + 0 = 1, not as the SUM of adding the numbers, but as the RESULT of counting the one item in the first group and then not having anything else to add to it, so ending up with a count of 1. So, e.g., the "answer" or result of 1 + 0 is defined not as a sum, but as a count, because the 0 blocks the operation of adding. It does ~not~ block the operation of counting. If we don't see any items, we say "0" ! I certainly agree with your subsequent comment that calling 0 a non-sum works out the same as calling it a sum. But that is consistent with my saying that I am merely trying to provide a correct metaphysical interpretation of what is going on with 0 that accords with common sense understanding of arithmetic, and which avoids the contradictory implications of comments like "adding 0 chairs to a room."]

Laure, you might have missed my reply to this example:

You say the ~sum~ is undefined. But you still say x = 12. You just don't proceed beyond the 12. It is as if ~nothing~ followed the 12. Or, I should say, it is because there ~isn't anything~ that follows the 12.

The example you gave would actually proceed, from left to right (foregoing the obvious arithmetic shortcut of reducing 14-14 to 0), as: 12 + 14 = 26 - 14 = 12. The virtue of combining 14-14 is that you actually, by visual inspection, are able to reduce the number of calculations/operations from two to one. Now you realize, by inspection, that you can disregard the +14 and -14, because their sum produces a non-operator, 0, and you can stop with the 12.

I am saying that the sum of 0 and x is = to the non-sum of x and 0.

There is nothing controversial in the result of an operation being equal in magnitude to a number on which no operation has been performed. 5 + 0 represents a number on which no operation is performed, and it is indeed equal to 0 + 5, the result of an operation. [Clarification on 5/5/13: this is true, but irrelevant, because actually neither 5 + 0 nor 0 + 5 is actually a possible addition operation, so there is no SUM in either case. However, there is a RESULT, namely, the count of items: in the first case 5 items without any other items being added to them, in the second case 5 items not being added to any other items. The count is the same: 5. But there isn't any addition that has been performed, since we weren't combining two groups of items in either case.]

Think "Occam's Razor", keep it simple.

Occam's Razor does not say "keep it simple." It says avoid ~unnecessary~ complexity, and as a corollary: avoid ~inadequate~ simplicity! I.e., don't unnecessarily complicate, and don't oversimplify! http://www.objectivistliving.com/forums/public/style_emoticons/#EMO_DIR#/poke.gif

REB

[Roger Bissell]

ATTENTION, ALL: this does NOT mean that I am endorsing additive commutativity of 0. The left hand of the equation represents a sum, but the right hand of the equation represents a non-sum. They just happen to be equal, because when you don't have anything and ~actively increase~ what you have by something, that gives the same result as when you have something and you ~don't actively increase~ it by anything. One is a sum, the other is not. That is why there is no such thing as additive commutativity of 0, even though the expressions 0 + x and x + 0 are equal.

REB

Roger. Let S be the set of (integers, rational numbers, real numbers, complex numbers). The binary operation addition (+) is a mapping from S x S into S. It happens to be commutative and associative. That is:

1. a + b = b + a

2. a + (b + c) = (a + B) + c

The statement 0 is commutative is meaningless.

0 is a number. It is a special number. It is the identity for the commutative (S, +), where S is defined as before.

Ba'al Chatzaf

I've heard of "fascism with a smiley face" (see Jonah Goldberg's excellent book, Liberal Fascism), but this is the first time I've seen ~mathematics~ with a smiley face. :poke:

REB

[Laure]

Roger, you can call x + 0 a non-sum if you feel like it, but my point is that there is absolutely no need to make a special case out of it. It works just like 0 + x; it doesn't need to be treated any differently.

[BaalChatzaf]

I've heard of "fascism with a smiley face" (see Jonah Goldberg's excellent book, Liberal Fascism), but this is the first time I've seen ~mathematics~ with a smiley face. :poke:

REB

Read the revised version. I killed the emoticons.

[merjet]

You compare adding zero to dividing by zero, when really what it's like is dividing by 1. What does it mean to "divide a number by 1"? You're NOT dividing it at all, right? But that doesn't mean that we say that x/1 is "undefined" - it's not, it equals x. Just like x + 0 is not "undefined", it's x.

I think Laure makes a good point here and Roger did not address it in his response (post #48). Nor does Roger's algorithm (also post #48) separately handle multiplying or dividing by 1. Roger, if adding 0 amounts to "doing nothing" in your view, then why doesn't multiplying or dividing by 1 amount to "doing nothing" in your view?

[Dragonfly]

1. Thom is correct. I do not propose to deny the commutative property of the ~multiplicative~ identity, 1.

But that is inconsequent, as the groups I mentioned are isomorphic, in other words, they have exactly the same structure, and that is what is essential in mathematics, not the particular example of that structure. So if you deny the communicative property in one example, you must also deny it in the other example. Or as Merlin said "if adding 0 amounts to "doing nothing" in your view, then why doesn't multiplying or dividing by 1 amount to "doing nothing" in your view?", which is an illustration of the isomorphism between the two groups in everyday words.

The essence of mathematics is that it is abstract and not dependent on any particular example to which it may be applied. Otherwise it would be like saying that the "3" in "3 horses" is mathematically different from the "3" in "3 unicorns" because horses do exist and unicorns do not. However, mathematics is not about horses, unicorns, chairs or anything physical, 0 is a mathematically defined number obeying the rules of mathematics, it is not the same as "nothing". An abelian group (A, •) is defined as a commutative group, that is, for all a,b in A: a • b = b • a, from which follows that also e • a = a • e, with e the unit element, which is 0 in my first example and 1 my second example. It is complete nonsense to declare the latter statement invalid by referring to some juggling (or not) with chairs and in fact declaring that abelian groups are not a valid concept (erase them from the mathematics handbooks!). The abelian group is a self-contained concept that is worth studying in itself without being pestered by metaphysical ramblings about chairs.

[merjet]

difference = minuend - subtrahend (if subtrahend not = 0)

else, difference is undefined

(There ~is no~ difference, because there is no operation. You can't subtract NOTHING from SOMETHING.)

P1: Wow, it sure warmed up this morning. How much did the temperature rise?

P2: It was 40 degrees earlier and now it's 65 degrees. 65 - 40 = 25, so 25 degrees.

A few days later ...

P1: Wow, it sure warmed up this morning. How much did the temperature rise?

P2: It was 0 degrees earlier and now it's 25 degrees. 25 - 0 = (pause) Er, I can't say because Mr. Bissell tells me I can't subtract 0 from something.

:console:

[Alfonso Jones]

difference = minuend - subtrahend (if subtrahend not = 0)

else, difference is undefined

(There ~is no~ difference, because there is no operation. You can't subtract NOTHING from SOMETHING.)

P1: Wow, it sure warmed up this morning. How much did the temperature rise?

P2: It was 40 degrees earlier and now it's 65 degrees. 65 - 40 = 25, so 25 degrees.

A few days later ...

P1: Wow, it sure warmed up this morning. How much did the temperature rise?

P2: It was 0 degrees earlier and now it's 25 degrees. 25 - 0 = (pause) Er, I can't say because Mr. Bissell tells me I can't subtract 0 from something.

:console:

There's hope, Merlin! If you just convert to Celsius, Roger will let you do the subtraction, because neither the 25 or the 0 in P2 is a zero. Then, you can convert back to Fahrenheit and get the answer.

Does anybody sense something is deeply wrong when the answer is not invariant under simple transformation of units - - - when an operation is "forbidden" with certain numbers in Fahrenheit, but not in Celsius?

Bill P (who is trying to contain laughter, but not succeeding)

[Roger Bissell]

difference = minuend - subtrahend (if subtrahend not = 0)

else, difference is undefined

(There ~is no~ difference, because there is no operation. You can't subtract NOTHING from SOMETHING.)

P1: Wow, it sure warmed up this morning. How much did the temperature rise?

P2: It was 40 degrees earlier and now it's 65 degrees. 65 - 40 = 25, so 25 degrees.

A few days later ...

P1: Wow, it sure warmed up this morning. How much did the temperature rise?

P2: It was 0 degrees earlier and now it's 25 degrees. 25 - 0 = (pause) Er, I can't say because Mr. Bissell tells me I can't subtract 0 from something.

:console:

I'm sorry, Merlin, I can't be pestered by your metaphysical ramblings about temperature.

Yours truly,

Flagondry

[Roger Bissell]

difference = minuend - subtrahend (if subtrahend not = 0)

else, difference is undefined

(There ~is no~ difference, because there is no operation. You can't subtract NOTHING from SOMETHING.)

P1: Wow, it sure warmed up this morning. How much did the temperature rise?

P2: It was 40 degrees earlier and now it's 65 degrees. 65 - 40 = 25, so 25 degrees.

A few days later ...

P1: Wow, it sure warmed up this morning. How much did the temperature rise?

P2: It was 0 degrees earlier and now it's 25 degrees. 25 - 0 = (pause) Er, I can't say because Mr. Bissell tells me I can't subtract 0 from something.

:console:

There's hope, Merlin! If you just convert to Celsius, Roger will let you do the subtraction, because neither the 25 or the 0 in P2 is a zero. Then, you can convert back to Fahrenheit and get the answer.

Does anybody sense something is deeply wrong when the answer is not invariant under simple transformation of units - - - when an operation is "forbidden" with certain numbers in Fahrenheit, but not in Celsius?

Bill P (who is trying to contain laughter, but not succeeding)

It's not as bad as all that. You don't need to convert to Celsius. That would just make it harder! You don't even need to (attempt) to perform subtraction by 0. At worst, you can start with the 0 and say (if you're ~really~ slow): well, if it warmed up 20 degrees, then it would now be 20 degrees. (0 + 20 = 20) Looks like it warmed up a little more than 20 degrees. 30 degrees? Naw, because then it would now be 30 degrees. (0 + 30 = 30) Ah, I see, it must have warmed up 25 degrees! (0 + 25 = 25)

Most people are actually intelligent enough to see, by inspection, how much it warmed up from 0 degrees to some higher temperature. They actually understand the question in terms of ~adding the higher temperature to 0 degrees~, not subtracting 0 degrees from the higher temperature. The real mental work comes in when you have to subtract a non-zero number from a higher number -- which is why I'd rather deal with calculating temperature rises from 0 degrees Fahrenheit than from its Celsius equivalent!

And how different from this is asking: the room had no chairs in it yesterday, and now it has five chairs, so how many chairs were added to the room? Do you ~really~ need to (try to) subtract zero chairs from five chairs? Come on!

It is humorous, Merlin and Bill, but so are other misguided attempts at ridicule. Have fun!

REB

[Roger Bissell]

I don't think we're going to get much further on this. It seems like both sides are fairly firmly dug in, and no one is scoring critical points -- and least none the other side wants to acknowledge!

So, I'm going to take a few days off from this thread and check out what Pat Corvini has to say about the concept of "zero" and how it functions (or doesn't) in arithmetic calculations. I don't recall from her lectures on the number system exactly what she said. She places a lot of emphasis on ~relations~ (conceptual and quantitative and numeral, not sexual!), which GS also referred to a few posts back.

I'll definitely get back to you all, if I rethink my position and see that I need to delete a bunch of posts! But that's just a nod to objectivity and non-dogmatism, not an indication that I have any doubts at this point.

Later,

REB

[tjohnson]

It is humorous, Merlin and Bill, but so are other misguided attempts at ridicule. Have fun!

Yes, ridicule doesn't accomplish much. I really believe that what Roger is talking about is something that everyone making the move from applied math to pure math has to deal with. From an applied math perspective it makes no sense to subtract 4 from 3, for example. When we teach children about counting they learn to make strong associations with numbers and "things" so the concept of subtracting more than you have makes no sense. In more formal, abstract math there actually is no operation called 'subtraction', there is only addition and multiplication. So, under addition for example, every element x has an additive inverse, -x, such that x + (-x) = 0. The conceptual leap comes in the realization that an operation in the system is nothing more than associating 2 elements with a one element. So;

(2,3) under addition = 5

(2,-1) under addition = 1

(-2,-3) under addition = -5

(2,0) under addition = 2

etc.

It is extremely important to remember that these numbers are not quantities of any objects, they are simply symbols that are associated together. Unfortunately, after many years of training our minds to think of numbers only has things to count and measure it is difficult to make this conceptual change.

Edited May 13, 2009 by general semanticist

[merjet]

I'm sorry, Merlin, I can't be pestered by your metaphysical ramblings about temperature.

Yours truly,

Flagondry

That was funny. Thanks for the laugh.

And how different from this is asking: the room had no chairs in it yesterday, and now it has five chairs, so how many chairs were added to the room? Do you ~really~ need to (try to) subtract zero chairs from five chairs? Come on!

It is humorous, Merlin and Bill, but so are other misguided attempts at ridicule. Have fun!

It is different in that temperatures are attributes and chairs are objects/entities. I believe General Semanticist is basically correct. Some things done for good reason in pure math don't make great sense in applying such math to some real world experience. His allusion to negative numbers in arithmetic is a good example. A room containing -5 chairs doesn't make sense in the real world. However, a temperature of -5 degrees, a net worth of -5 dollars, or a rate of change of -5 miles per hour does make sense in the real world. Note that all three are attributes, not entities.

It may feel like ridicule to you, but my comments have not been intended as ridicule. I have tried to point to some odd consequences of your position. You have been a good sport about it. I don't consider you an ignoramus, but that your views are peculiar or idiosyncratic or outdated. Some prominent past mathematicians considered negative numbers fictitious. Even Decartes, as great a mathematician as he was, considered negative solutions to quadratic equations false roots (link). And we haven't yet delved into imaginary numbers!

You have given respectable reasons for your position, but I don't judge them as strong enough reasons to hold them myself. I doubt it is all that simple, but maybe it is a matter of your trying to maintain a 'counting entities' perspective of numbers, and adding an 'attribute' perspective of numbers will alter your thinking.

I don't agree with General Semanticist, who has said "Mathematics makes no claims about existence whatsoever" (post #7 ). My position is that some of mathematics makes no claims about reality. On the other hand, much of mathematics is very much applicable to, and rooted in, reality.

I will be posting little or none for the next several days due to travel.

I would like to hear Pat Corvini's lectures, but ARI's prices are too high for me.

Edited May 13, 2009 by Merlin Jetton

[tjohnson]

I don't agree with General Semanticist, who has said "Mathematics makes no claims about existence whatsoever" (post #7 ). My position is that some of mathematics makes no claims about reality. On the other hand, much of mathematics is very much applicable to, and rooted in, reality.

Pure mathematics makes no claims about existence whatsoever, not applied mathematics. Physics, for example, is largely applied mathematics and of course it is very much about "reality'.

[merjet]

Pure mathematics makes no claims about existence whatsoever, not applied mathematics. Physics, for example, is largely applied mathematics and of course it is very much about "reality'.

But what is the demarcation between pure and applied math? I can accept the one given here, which is based on intention. However, with respect to reality, the content of pure math can also be content for applied math.

Edit: If a reader is interested in more about this, the Wikipedia articles on pure math and applied math look okay. Again, the distinction is based on intention.

Edited May 13, 2009 by Merlin Jetton

[tjohnson]

OK, here is an example. In pure mathematics the series 1/2, 1/4, 1/8...,1/2n never reaches zero. If you try this "in reality", like halving the distance between you and some line, you will effectively reach it. Mathematics deals with exact relations, which don't exist in "reality'. I think it was Einstein that said "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality".

[BaalChatzaf]

Pure mathematics makes no claims about existence whatsoever, not applied mathematics. Physics, for example, is largely applied mathematics and of course it is very much about "reality'.

But what is the demarcation between pure and applied math? I can accept the one given here, which is based on intention. However, with respect to reality, the content of pure math can also be content for applied math.

Edit: If a reader is interested in more about this, the Wikipedia articles on pure math and applied math look okay. Again, the distinction is based on intention.

Applied mathematics is a tool. Pure mathematics is art. As you say, this pertains more to intent than content. The content of applied (or applicable mathematics) can scarcely be distinguished from the content of pure mathematics. In fact what is now applicable mathematics was once thought (or intended) to be pure mathematics. For example the theory of Galois Groups and finite fields was once considered to be very pure math. Now both find uses in coding theory. Or consider geometry: From Riemannian Geometry -> Theory of General Relativity (gravity) -> GPS. You can't get much more practical than GPS. Just about every graduate text in mechanics dwells on manifolds and fiber bundles. Hoodahthunkit? My speculation is that every field of pure mathematics currently so designated will some day find a practical use.

Ba'al Chatzaf

[BaalChatzaf]

I think you are right. Your presentation of the misinterpretations of the philosophy behind mathematics in this context deserves serious study. If we take the Aristotelian-Objectivist approach as the correct line of sight, then Objectivists have much work ahead of us to turn the ship of Math toward the right direction.

If the Objectivists get a hold of mathematics (Reality forbid!) they will turn it right into the rocks. Some O'ists got a hold of physics and produced the Theory(???) of Elementary Waves; pure crackpot and crank nonsense. I shudder to think what they will do to math.

Progress in mathematics went at warp speed once it was liberated from metaphysics.

Ba'al Chatzaf

Ba'al, I'm still waiting to hear how denying the commutativity of zero in abelian groups will cause the catastrophic meltdown of mathematics.

In the meantime, I would like two or three concrete historical examples of rapid advances in mathematics that were made possible by the jettisoning of metaphysics. Please pick at least two that you think are decisive illustrations of your point.

Thanks in advance.

REB

For the n-th time. Commutativity is a property of binary operators, not of individual elements. Consider an operation ^: S x S -> S. ^ is commutative if and only if ^(s1, s2) = ^(s2, 21). If we write this in the manner of an infix operator we get the more familiar form:

s1 ^ s2 = s2 ^ s1.

Riemannian Geometry. Klein's Erlanger Programme in Geometry. Euclidean Geometry is now rendered as the set of invariants under isometry. In algebra, the theory of fields and ideals separates the algebraic properties of number systems from their quantitative connotations. In all these cases the process of abstraction has moved the mathematics its historical origins of how many and how much to a new level of abstraction. The mathematics is now about form and structure, rather than quantity. The all time champ of removing mathematics from the clutches of philosophy is Category Theory.

Hilbert's program of regarding mathematics as a formalism, as opposed to statements about the world or even about Platonic Ideas. Hilbert's program of complete reducing mathematics to formalism was stopped cold by the Goedel Incompleteness theorems. Pure formal provability, it turns out, cannot capture mathematical truth. Yet much of Hilbert's program is the basis of applying computers to such tasks as showing proofs are valid. It turns out validity proving is algorithmic.

And of course making number theory, the purest of the pure Platonic mathematical pursuits to be a tool in the kit of cryptographers. All modern systems of ciphers are dependent on number theory or group theory. The Enigma Cipher was initially dented (but not cracked) by the Polish mathematician Rejewsky using group theory, shortly before the outbreak of WW2 in 1939. The Brits picked up what Rejewsky started and you have the famous break through by Turing when he worked at Bletchley Park.

Ba'al Chatzaf

Edited May 13, 2009 by BaalChatzaf

[Christopher]

I can offer a psychological perspective:

mathematics can operate through two independent brain systems. There is the "applied" part and the "abstract" part. Applied and abstract mathematics are handled by different cognitive areas.

Applied is "20 of 100 people."

Abstract is "20% of 100 people"

If you ask the average college-educated individual which of the two quantities above is greater, they will laugh and tell you that the quantities are the same. However, research has demonstrated that if you ask the average non-educated American (who is not very good at abstraction) which of these quantities is greater, more often the answer is not that they are equal. Just by adding the "%" sign, people get this wrong!

This and other evidence suggests that mathematical operations such as percentages (and probably fractions, etc) are handled by an abstract part of the brain; whereas, quantities such as "taking 3 oranges out of 5 oranges" are handled by a more concrete-thinking part of the brain.

So now when we're dealing with abstract mathematics or concrete quantities within reality, different areas of the brain are activating. Therefore, to compare mathematical operations with concrete qualities of reality is like comparing apples to oranges or sight to sound.

Chris

[tjohnson]

Very good perspective Chris. I have some experience teaching mathematics and getting people to use the abstract part of the brain is very difficult at first. This is why there is such widespread dislike for mathematics I suspect. Most people when presented with very abstract formulations will respond by saying "what use will this ever be?" or something and, in most cases, probably it will be of no use in practice. But there is another reason to learn mathematics besides applying it to engineering and science and that is because it represent the highest level of human thinking ever achieved and can provide a basis for critical thinking in general.