The Conspiracy of Mathematics

[gwnn]

divide by (B-A)

does not compute

does not compute

does not compute

[Winstonm]

Pretty basic

 

does not compute

does not compute

does not compute

[mike777]

The one scene in the movie "A Beautiful Mind" that stuck with me was where Russell Crowe hated to teach; he just hated it. He thought his students were a waste of his time.

 

--

 

In studying for (one level out of 3) my CFA exam I remember meeting a math prof who said the exam was really tough he had studied for 6 weeks. I had studied for almost a year 7 days aweek...Math guys are really good. :)

[kenberg]

The one scene in the movie "A Beautiful Mind" that stuck with me was where Russell Crowe hated to teach; he just hated it. He thought his students were a waste of his time.

 

--

 

In studying for (one level out of 3) my CFA exam I remember meeting a math prof who said the exam was really tough he had studied for 6 weeks. I had studied for almost a year 7 days aweek...Math guys are really good. :)

 

Yeah, and our social skills are really top stuff also.

 

 

When I was a grad student one of my fellow students would frequently approach me with the question "Do you know how dumb I am?" He would then continue "I didn't even know..." and then download everything he had learned in the last day or so.

 

Basically I like my fellow mathematicians but they can be odd. My wife recently noted that one of the guys that we have known for the 16 years that we have been married recently actually referred to her as "Becky" rather than "She".

[Winstonm]

Math is to Winston as sober is to curling.

[mike777]

fwiw as a student in high school I just remember how easy math was once you get it and how bad the teacher was in teaching it.

 

This seems even more true in college.

 

The teachers knew the subject but they did not know how to teach it. Which leads me in a circle back to how much they really knew the subject.

 

 

Lets teach transfers. You are an expert in transfers. Now teach it.

 

--

 

 

As far as the teachers I guess at the univ of chicago the saying was...we teach to the future nobel prize winners...we care less about the rest.

 

---

 

 

I fully grant many students could care less.

[kenberg]

Mathematicians don't get Nobel prizes. Unless they can convince someone they are economists or something. They can get a Fields medal.

[kenberg]

fwiw as a student in high school I just remember how easy math was once you get it and how bad the teacher was in teaching it.

 

This seems even more true in college.

 

The teachers knew the subject but they did not know how to teach it. Which leads me in a circle back to how much they really knew the subject.

 

[/Quote]

 

I have heard all of my life about how the really good mathematicians can't teach, and I want to respond. On occasion that may be true but mostly it's bunk. For example, Hillel Furstenberg was teaching at Minnesota when I was a grad student there. He soon moved to Hebrew University and is one of the top people in a very top level school. I had to take a required course in philosophy at the hour that he was teaching a different course, but I only got to the required course on Jewish holidays since I far preferred to listen to Furstenberg. (I got an A, philosophy not being one of your more difficult subjects.) Charlie Fefferman got his Ph.D. from Princeton when he was 20 or so, maybe 21, and soon became a professor there. He has one of these Fields medals I mentioned. He is a clear and fascinating lecturer. Steve Smale, another Fields medalist, you will get mixed opinions about but I think he is a great lecturer. He gives the key ideas without much detail, asks if anyone can see anything wrong with his argument, and if no one speaks up he moves on to the next big idea. Definitely leaves room for filling in the gaps, but the key ideas he mentions really are the key ideas. These are three of many many examples. The most obvious difference between high school and college is the depth of knowledge possessed by the person in front of the class. To each his own, but I far preferred college to high school.

[mike777]

fwiw I will grant having a "tiger mom" would have helped me in all of my subjects but esp math.

 

Many of us have something close to we hope mom can pay the rent mom; even today.

 

 

 

have heard all of my life about how the really good mathematicians can't teach

[wyman]

have heard all of my life about how the really good mathematicians can't teach

Mostly this is a function of social aptitude. The most knowledgeable people about an area will by-and-large be the best teachers; the problem is that there are an awful lot of socially awkward and inept people in the hard sciences.

 

The other thing is that good teachers make hard things appear easy. The downside of this is that the teacher appears less smart in contrast to his colleague who babbles about some seemingly impossible concept (which would be possible if the right person were explaining).

[kenberg]

There is something to this last comment. I think some colloquium speakers are terrified by the thought that someone will say that they found the talk really easy to follow.

[wyman]

There is something to this last comment. I think some colloquium speakers are terrified by the thought that someone will say that they found the talk really easy to follow.

 

This is a huge concern when giving job talks. You don't want the audience saying "yeah, but that was kind of trivial."

[kenberg]

This is a huge concern when giving job talks. You don't want the audience saying "yeah, but that was kind of trivial."

 

A job is good, tenure is better, retirement is really where it's at.

[shyams]

Let me tell you something: everything you've ever known about math is wrong. 1 + 1 = ??? Wrong, 1 + 1 = 1, not 2!

 

Suppose A = B = 1.

 

multiply with A

 

A*A = A*B

 

multiply with -1

 

-(A*A) = -(A*B )

 

add B*B

 

B*B - A*A = B*B - A*B

 

rearrange

 

(B-A)*(B+A) = (B-A)*B

 

divide by (B-A)

 

(B-A)/(B-A) * (B+A) = B

 

B + A = B

 

1 + 1 = 1

 

 

does not compute

does not compute

does not compute

Here's another one -- again easy to find the flaw.

 

We start by saying: -20 = -20

This is the same as: 16 - 36 = 25 - 45

Add in (81/4): 16 - 36 + 81/4 = 25 - 45 + 81/4

Re-state as: (4 + 9/2)^2 = (5 + 9/2)^2

Take sq roots: 4 + 9/2 = 5 + 9/2

Eliminate 9/2 from both: 4 = 5

[BunnyGo]

Here's another one -- again easy to find the flaw.

 

We start by saying: -20 = -20

This is the same as: 16 - 36 = 25 - 45

Add in (81/4): 16 - 36 + 81/4 = 25 - 45 + 81/4

Re-state as: (4 + 9/2)^2 = (5 + 9/2)^2

Take sq roots: 4 + 9/2 = 5 + 9/2

Eliminate 9/2 from both: 4 = 5

 

Do the factorization wrong? I think you meant (4 - 9/2)^2 and (5 - 9/2)^2.

 

But the essential flaw is very cute. I hadn't seen this one presented this way before.

[kenberg]

These things can be subtle. Some 45 years ago when I was writing my thesis I cam up with a method that helped me prove a result I believed to be true. Then I noticed that this method also quickly established another result. And then another. Then I got suspicious. Then I found my error. It was more subtle than the product of (-2)X(-3). Same general idea though: Stay calm, think it through, then you get it right.

[Winstonm]

It should be a shock to no one that I bareley made it though algebra with my skin intact, so I studied zero mathematical theory. But I have a question.

 

It appears to me that there are a number of apparently reasonable ways to think about the basic problem presented of (-2)x(-3), including Ken's correct method of removing two -3 debits to create +6 increase in wealth, but it also seems to me just as valid with this same method that Ken used to come to the conclusion that by eliminating the debits, you have arrived at zero, as you had to be at -6 to start in order to have two -3 debts in the first place.

 

Now, it does make sense from an accounts payable basis, that if business x pays business y by transfering two -3 debts due that business y has gained +6, but why wouldn't that be expressed as 2x(-3)? for business x, and (-2)x3 for business y?

 

So, to my question - is it pure reasoning that leads to the right method of thinking about the equation, or is correct thinking a product of axiomatic stipulations?

 

Thanks to anyone with the patience to answer.

[Echognome]

Winston - How about this? You have two line items under accounts payable of $3 each. You then remove them with (-2)x(-$3). What happens to your ending balance?

[wyman]

Yes, Winston, if you're going to be thinking about numbers as being debts you owe and debts you are owed, you should think about the numbers not as absolutes; rather you should consider them as operators on your total balance.

 

So (-2) x (-3) is the number which operates by removing 2 debts of 3 dollars from your balance. That is, your balance will go from B to B+6. (this is the same effect as if we had just received a +6 to begin with; hence these are the same "number" or "operator on balance")

 

To someone's other point, that this clearing of debt should net 0. Your reasoning is slightly bad, since you may have started not with a debt of 6 but with a debt of 100. This only serves to show that -2x-3 needn't be 0 (that is, that the logic by which someone arrived there is bad). That -2 x -3 actually equals 6 I claim is evident by the above.

 

-----------------

 

Also, if we believe that:

(a) multiplication of whole numbers commutes (that is, a x b = b x a for every pair a, b );

(b )multiplication is associative (order does not matter: (a x b ) x c = a x (b x c))

(c )-1 x a = -a for every positive a

(d) -1 x -1 = 1

 

then we can show that -2 x -3 = 6 formally, without considering numbers as debts or balances.

 

Then, -2 x -3 = (-1 x 2) x (-1 x 3) = [since (a) and (b ) allow us to regroup and swap as we wish]

(-1 x -1) x (2 x 3) =

1 x 6 =

6

 

At the very least, we've reduced the problem to convincing ourselves that -1 x -1 = 1.

 

edit: freakin' smilies

[Winstonm]

(a) multiplication of whole numbers commutes (that is, a x b = b x a for every pair a, b );

(b )multiplication is associative (order does not matter: (a x b ) x c = a x (b x c))

(c )-1 x a = -a for every positive a

(d) -1 x -1 = 1

 

then we can show that -2 x -3 = 6 formally, without considering numbers as debts or balances.

 

Then, -2 x -3 = (-1 x 2) x (-1 x 3) = [since (a) and (b ) allow us to regroup and swap as we wish]

(-1 x -1) x (2 x 3) =

1 x 6 =

6

 

Wyman,

 

Thanks for the explanation - this last makes sense to me, but I note that it is axiomatically-driven.

 

In other words, to me it appears that the correct answer cannot be - for want of a better word - reasoned out without knowledge of the axiomatic bases. Or perhaps it can be, but it is easier to come to incorrect conclusions if you try to "reason it out" instead of following the logical path provide by the axioms.

[Winstonm]

Winston - How about this? You have two line items under accounts payable of $3 each. You then remove them with (-2)x(-$3). What happens to your ending balance?

 

The ending balance is improved by +6. Let's see, my accounts payable is a negative to my bottom line, so I have two -3's. (O.K. so far.) I remove them...

 

Hmmm. That language gets a bit subjective. How do I remove them - cook the books and simply erase them, or take in +6 from accounts receivable to negate them? If I negate them with income from accounts receivable, my net profit is zero on that transaction. It is only a +6 profit if my accountant can "make them disappear", wink, wink, nudge, nudge.

 

I'm kidding - kind of. I see what you are saying, that if the two -3's are eliminated I have gained +6 to my bottom line. At the same time, note the confusion from simply attempting to grasp what exactly is meant by "taking away".

[Echognome]

Winston - The "taking away" part is easy. We suppose that the parties that issued us invoices (and triggered the accounts payable) have canceled those invoices. I don't think you can extend anything to accounts receivable, because those are derived from different underlying transactions (such as our company issuing invoices to a completely different party). We are talking only about a change in AP, not a change in AR.

 

So let's take this example to your specific questions and continue our hypothetical.

 

Q: Do I remove them by taking in +6 from accounts receivable?

A: No. This was a cancellation of an invoice issued to our company and does not affect our AR. The party that canceled the invoices is one of our vendors and therefore we do not issue invoices to this party. We have no AR with this vendor at all, so there is nothing to net or offset.

 

Q: If I negate them with income from accounts receivable, my net profit on that transaction is zero, right?

A: No. They are from completely separate transactions. In addition, any discussion of "profit" should be referred to my profit and loss statement rather than my balance sheet. The only netting that occurs is when I am determining what my current balance is.

 

Edit: I just thought of a way to summarize. We state the question as:

 

"What happens to our balance sheet when a vendor cancels two $3 invoices issued to us for payment?"

 

Answer: Our balance sheet gains $6 as (-2)x(-$3) = $6.

[BunnyGo]

Wyman,

 

Thanks for the explanation - this last makes sense to me, but I note that it is axiomatically-driven.

 

In other words, to me it appears that the correct answer cannot be - for want of a better word - reasoned out without knowledge of the axiomatic bases. Or perhaps it can be, but it is easier to come to incorrect conclusions if you try to "reason it out" instead of following the logical path provide by the axioms.

 

You suggest that the axioms are not "reasonable" (something which can be "reasoned out") in that you have been mentioning them in contrast to each other. Isn't it the case that a student does (or can if given the chance) reason out these "rules" (in fact, isn't that how people found them in the first place?). The axioms of arithmetic are simply what many people have found reasonable and "discovered" and so wrote them down. Likewise the "laws" of physics are simple patterns and mathematics that people have found reasonable.

 

One reasons out that 2+3 = 3+2 or that 2*3 = 3*2. After enough examples where this pattern emerges, one may find their own reason for it: My favorite is that 2+2+2 = 3+3 by taking one from each 2 for the first 3, and one from each 2 for the second 3.

 

At the end of the day, I think any explanation which makes sense to you is just as valid and reasoned out as another.

 

I also think wyman usually explains these things better than I do...

[Winstonm]

Winston - The "taking away" part is easy. We suppose that the parties that issued us invoices (and triggered the accounts payable) have canceled those invoices. I don't think you can extend anything to accounts receivable, because those are derived from different underlying transactions (such as our company issuing invoices to a completely different party). We are talking only about a change in AP, not a change in AR.

 

So let's take this example to your specific questions and continue our hypothetical.

 

Q: Do I remove them by taking in +6 from accounts receivable?

A: No. This was a cancellation of an invoice issued to our company and does not affect our AR. The party that canceled the invoices is one of our vendors and therefore we do not issue invoices to this party. We have no AR with this vendor at all, so there is nothing to net or offset.

 

Q: If I negate them with income from accounts receivable, my net profit on that transaction is zero, right?

A: No. They are from completely separate transactions. In addition, any discussion of "profit" should be referred to my profit and loss statement rather than my balance sheet. The only netting that occurs is when I am determining what my current balance is.

 

Edit: I just thought of a way to summarize. We state the question as:

 

"What happens to our balance sheet when a vendor cancels two $3 invoices issued to us for payment?"

 

Answer: Our balance sheet gains $6 as (-2)x(-$3) = $6.

 

 

O.K. If I have 6 apples set aside to pay back my apple-debt to Bill Jones, and Bill Jones cancels two 3-apple debts, then those 6 apples are now mine again, and I have a positive 6 apples.

 

The key here seems to be remembering (if a child) that the (-2) of the (-)x(-) equation represents the number of times an elimination of debt occurs while the (-3) represents the amount of each debt.

[Winstonm]

You suggest that the axioms are not "reasonable" (something which can be "reasoned out") in that you have been mentioning them in contrast to each other. Isn't it the case that a student does (or can if given the chance) reason out these "rules" (in fact, isn't that how people found them in the first place?). The axioms of arithmetic are simply what many people have found reasonable and "discovered" and so wrote them down. Likewise the "laws" of physics are simple patterns and mathematics that people have found reasonable.

 

One reasons out that 2+3 = 3+2 or that 2*3 = 3*2. After enough examples where this pattern emerges, one may find their own reason for it: My favorite is that 2+2+2 = 3+3 by taking one from each 2 for the first 3, and one from each 2 for the second 3.

 

At the end of the day, I think any explanation which makes sense to you is just as valid and reasoned out as another.

 

I also think wyman usually explains these things better than I do...

 

 

I did not mean to imply that rules are not reasonable. I only meant to say that if one relies on trying to figure out for oneself what seems to make sense, then there are many ways to go wrong - it is indeed logical and reasonable to follow the axiomatic pathway.