Counter-intuitive phenomena in math and physics

[jedson303]

I found the http://gowers.wordpress.com/2012/11/20/what-maths-a-level-doesnt-necessarily-give-you/ interesting. Curious that his example was the same one I was asking about. He makes the distinction between teaching how to understand math and teaching mechanical manipulations (my cheap tricks). I can share at least one student's reaction to this distinction. When I was taking high school algebra we started coming to some things that didn't make immediate intuitive sense to me. The first one was that a minus times a minus equaled a plus. In one way I could understand this, yet it seemed to me that if the minus and the plus areas on a number line were symmetrical extensions from 0, then the way it worked should be symmetrical on each side of 0. (If this makes sense.) I asked about this and was told that I needed to simply accept it and do the problems. As other things came up that also weren't intuitive to me, things that I was just to accept, I turned off on the whole process and stopped making an effort to learn algebra. A very similar thing happened with Physics. It had to do with the speed of light being the same as measured from Earth regardless of the speed at which different sources of the light were traveling in relation to the earth. This made no sense whatsoever to me. It was impossible, and I told the physics teacher it was. He said that it simply was true. I argued with him. It wasn't true of anything else, I pointed out. I was eventually told I just had to accept it. And at that point I turned off on physics. Much later I learned about the special theory of relativity -- and then about the general theory and then quantum physics, etc. This was fascinating stuff, which I could more or less grasp intuitively, but not mathematically. I regretted my earlier turning off on math and some 50 years after those fateful high-school courses, I took a pre-calculus and then a calculus course. They have hardly made me a math wizard. But learning this this math does promise to give me the tools for a better understanding of some fascinating stuff.

 

[Ackbach]

Re: Cheap tricks

OK, Opaig. I see how that works. It does connect. Thanks.


I found the Gower's weblog interesting. Curious that his example was the same one I was asking about. He makes the distinction between teaching how to understand math and teaching mechanical manipulations (my cheap tricks). I can share at least one student's reaction to this distinction. When I was taking high school algebra we started coming to some things that didn't make immediate intuitive sense to me. The first one was that a minus times a minus equaled a plus. In one way I could understand this, yet it seemed to me that if the minus and the plus areas on a number line were symmetrical extensions from 0, then the way it worked should be symmetrical on each side of 0. (If this makes sense.) I asked about this and was told that I needed to simply accept it and do the problems. As other things came up that also weren't intuitive to me, things that I was just to accept, I turned off on the whole process and stopped making an effort to learn algebra. A very similar thing happened with Physics. It had to do with the speed of light being the same as measured from Earth regardless of the speed at which different sources of the light were traveling in relation to the earth. This made no sense whatsoever to me. It was impossible, and I told the physics teacher it was. He said that it simply was true. I argued with him. It wasn't true of anything else, I pointed out. I was eventually told I just had to accept it. And at that point I turned off on physics. Much later I learned about the special theory of relativity -- and then about the general theory and then quantum physics, etc. This was fascinating stuff, which I could more or less grasp intuitively, but not mathematically. I regretted my earlier turning off on math and some 50 years after those fateful high-school courses, I took a pre-calculus and then a calculus course. They have hardly made me a math wizard. But learning this this math does promise to give me the tools for a better understanding of some fascinating stuff.

I would have handled those two particular examples differently, the one from the other. That a negative times a negative equals a positive comes from what a negative is. The negative of a number is the thing you must add to the number in order to get zero. So the negative of $5$ is $-5$, because $-5+5=0$. So what is the negative of a negative number? It's the thing you must add to the negative number to get zero. What must I add to $-5$ in order to get zero? Answer: $5$. Therefore, $-(-5)=5$. So there is a logical explanation of this property in terms of more fundamental concepts.

However, that the speed of light is independent of the velocity of its source is one of the assumptions you must make in special relativity. So the "why" question is answered as, "It's an axiom or postulate, like the Euclidean idea that through any point not on a given straight line is only one line that does not intersect with the given line." You must assume the speed of light postulate. Now, if someone asks why you assume the speed of light postulate, I would reply that doing so yields predictions that are closer to the outcomes of experiments than other conflicting assumptions (such as Galilean relativity).

My two cents.

 

[Opalg]

Re: Cheap tricks

However, that the speed of light is independent of the velocity of its source is one of the assumptions you must make in special relativity. So the "why" question is answered as, "It's an axiom or postulate, like the Euclidean idea that through any point not on a given straight line is only one line that does not intersect with the given line." You must assume the speed of light postulate. Now, if someone asks why you assume the speed of light postulate, I would reply that doing so yields predictions that are closer to the outcomes of experiments than other conflicting assumptions (such as Galilean relativity).

I am not a physicist, but I would put that a bit differently. Physics is different from mathematics in that it is founded on experiment. When the experiments of Michelson and Morley in the 1880s showed that there was no measurable effect on the speed of light by the velocity of its source, it must have seemed as mystifying to the scientific community as it did to the OP. It was only when Einstein put this peculiar phenomenon into a coherent theoretical framework that people were able to feel comfortable about it.

 

[topsquark]

Re: Cheap tricks

I am not a physicist, but I would put that a bit differently. Physics is different from mathematics in that it is founded on experiment. When the experiments of Michelson and Morley in the 1880s showed that there was no measurable effect on the speed of light by the velocity of its source, it must have seemed as mystifying to the scientific community as it did to the OP. It was only when Einstein put this peculiar phenomenon into a coherent theoretical framework that people were able to feel comfortable about it.

A quick comment from the peanut gallery about Special Relativity. There was a severe problem when the Maxwell equations were written down: They do not obey Galilean relativity...that is velocities do not add. Lorentz took up the problem and stated that EM obeys a different set of transformations where again the velocities did not add.

Einstein's genius was to assume the Lorentz equations also held for Mechanics (and everything else!) Great work, but no one seems to remember Lorentz for his contribution.

-Dan

 

[Opalg]

Re: Cheap tricks

A quick comment from the peanut gallery about Special Relativity. There was a severe problem when the Maxwell equations were written down: They do not obey Galilean relativity...that is velocities do not add. Lorentz took up the problem and stated that EM obeys a different set of transformations where again the velocities did not add.

Einstein's genius was to assume the Lorentz equations also held for Mechanics (and everything else!) Great work, but no one seems to remember Lorentz for his contribution.

-Dan

I am sure that Lorentz played a vital part in paving the way for Einstein, and I have even more admiration for the genius of James Clark Maxwell (Scotland's greatest-ever physicist, in my opinion) in daring to formulate equations that were inconsistent with the classical understanding of how velocities combine.

 

[MarkFL]

Re: Cheap tricks

I see Maxwell as having made the first significant step in the unification of the forces of nature. I truly hope to see this project completed in my lifetime.

 

[topsquark]

Re: Cheap tricks

I see Maxwell as having made the first significant step in the unification of the forces of nature. I truly hope to see this project completed in my lifetime.

Well we've got electo-weak theory (EM + weak nuclear force) reasonably pinned down now we've at least got a good idea about Strong-electroweak. Well, okay, there are several of these. But they all share many common traits at least.

So much for the GUT's. As far as getting gravity in there, again there are a number of theories that are in the running. And we still don't have a good theory of quantum gravity so we're kind of stymied when trying for a TOE. Gravity is going to take a while.

Although Doctor Michio Kaku and some others are acting like string theory is the answer. However they are still ignoring that we don't have any data to support it...

-Dan

 

[topsquark]

Re: Cheap tricks

Ummmm... I just realized I managed to hi-jack the thread. Sorry about that!

-Dan

 

[MarkFL]

Re: Cheap tricks

Ummmm... I just realized I managed to hi-jack the thread. Sorry about that!

-Dan

I don't think you've hi-jacked the topic...the OP has shown satisfaction with the replies given to the original query, and indicated an interest in modern physics, and so the discussion naturally evolved. (Nod)

edit: I did split this discussion into a separate topic, just so the discussion may continue without fear of derailing the original topic.