I recently read "A Mathematician's Apology", by G. H. Hardy.

Check this excerpt from Wikipedia's article on that book:

For Hardy, the most beautiful mathematics was that which had no practical applications in the outside world (pure mathematics) and, in particular, his own special field of number theory. Hardy contends that if useful knowledge is defined as knowledge which is likely to contribute to the material comfort of mankind in the near future (if not right now), so that mere intellectual satisfaction is irrelevant, then the great bulk of higher mathematics is useless. He justifies the pursuit of pure mathematics with the argument that its very "uselessness" on the whole meant that it could not be misused to cause harm. On the other hand, Hardy denigrates much of the applied mathematics as either being "trivial", "ugly", or "dull", and contrasts it with "real mathematics", which is how he ranks the higher, pure mathematics.

 

I'm having trouble understanding the request.

No offence meant , this is a serious request.

What do you understand by a Field?

The mathematical definition is certainly very different from the physics one.

That would be a good example.

 

Borrowing a concept from Isaac Asimov…

The numbers 1 and 2 are well understood, but how about 1 over 2, or one half?

Can you hand me half a glass of water?

In theory, yes.

In practice, absolutely not, not without adding some additional definition (the glass holds 8 ounces and I gave you half of that or 4 ounces) constraint (I am within .1% of the nominal).

Still you end up with one glass of water with some arbitrary quantity of water within.

Where is the half?

 

Where is the half?

It's definitely a conspiracy made by the same people that implemented daylight savings time... where did that hour of my life go?

 

But
, here's the first thing that popped into my head: Pi. It seems to me to be a failure of math that something so physically simple - the ratio of a circle's circumference to its diameter - is a never-ending number.

Why is that a failure? In fact, I'd think that if it weren't a never ending number, it would be as suspect. Take the set of real numbers. Remove all integers. You have an infinite number of numbers left. Repeat with numbers with one digit after the decimal point. An infinite set again. Repeat with n number of digits after the decimal point, where n = 2 to infinity. An infinite set. What's left? An infinite number of never - ending numbers. The fact that Pi and e, etc... are never ending comes as no surprise to me.

It's definitely a conspiracy made by the same people that implemented daylight savings time... where did that hour of my life go?

Only the federal government believes that cutting a strip from one end of a blanket and seeing it to the other end, makes a longer blanket - Native American observation.

 

Hi, erniem.

Nice, footwork I like it.

Half a hole is even better.

 

I recently read "A Mathematician's Apology", by G. H. Hardy.

Check this excerpt from Wikipedia's article on that book:

For Hardy, the most beautiful mathematics was that which had no practical applications in the outside world (pure mathematics) and, in particular, his own special field of number theory. Hardy contends that if useful knowledge is defined as knowledge which is likely to contribute to the material comfort of mankind in the near future (if not right now), so that mere intellectual satisfaction is irrelevant, then the great bulk of higher mathematics is useless. He justifies the pursuit of pure mathematics with the argument that its very "uselessness" on the whole meant that it could not be misused to cause harm. On the other hand, Hardy denigrates much of the applied mathematics as either being "trivial", "ugly", or "dull", and contrasts it with "real mathematics", which is how he ranks the higher, pure mathematics.

Frankly, this just strikes me as someone trying to find a justification for setting themselves up on a pedestal and looking down on everyone else. Someone that spends their time arranging seashells on the beach in the tidal zone could come up with a very similar sounding reason for why their life is "pure" and everyone else's life is "trivial".

 

I don't like to sound harsh, but Hardy is not counted amongst the mathematics greats of his day, such as David Hilbert.
Although Professor of Mathematics at Cambridge and author of 'A Course in Pure Mathematics', he did not produce much original new work.
His work contained many small, often obscure, insights and quibbles, in the fashion of lawyers.

 

I don't like to sound harsh, but Hardy is not counted amongst the mathematics greats of his day, such as David Hilbert.
Although Professor of Mathematics at Cambridge and author of 'A Course in Pure Mathematics', he did not produce much original new work.
His work contained many small, often obscure, insights and quibbles, in the fashion of lawyers.

I've never heard of him and I would have made the same commentary had the quote come from anyone in any field. But what you say definitely jives with my experience, namely that people that fall into the category you describe are often exactly the people that need to justify their "greatness" at others' expense.

 

He himself said, his main claim to fame was that he discovered Ramanujan, (albeit too late) and sponsored him to England.
He was certainly prepared to yield to Ramanujan's greatness, sadly cut so short, but developed so well by Emmy Noether.

But that has been the history of several greats of pure mathematics.

Galois, and Abel both produced masterwork before being cut down in their youth.

 

He himself said, his main claim to fame was that he discovered Ramanujan, (albeit too late) and sponsored him to England.
He was certainly prepared to yield to Ramanujan's greatness, sadly cut so short, but developed so well by Emmy Noether.

But that has been the history of several greats of pure mathematics.

Galois, and Abel both produced masterwork before being cut down in their youth.

You're right about Mr Hardy's arrogance... and that possibly his greatest achievement was not turning Ramanujan down, like some of his own peers did, before discovering what a jewel he was.
As for a mathematician's life being cut short, sadly the greatest work in mathematics is normally done when the researcher is younger than 30 (or even less)... after that, creativity and innovation (in the field of mathematics) decline.

 

As for a mathematician's life being cut short, sadly the greatest work in mathematics is normally done when the researcher is younger than 30 (or even less)... after that, creativity and innovation (in the field of mathematics) decline.

What evidence do you have for this statement and what % of mathematicians creating all before 30 would you class as 'normal'?

 

What evidence do you have for this statement and what % of mathematicians creating all before 30 would you class as 'normal'?

Evidence? Only anecdotal, unfortunately ... though maybe I could find a few examples later on. Also, I didn't mean to say that all of their important stuff was made before 30, I meant that most of it is done before that age, and that creative decline follows afterwards.
As for "normal" mathematicians... maybe that's an oxymoron, since one has to be pretty exceptional to venture into that field in the first place... What's normal is that geniuses are also eccentrics . But that's only my humble opinion, of course.

 

I know a long decline like a damped sinusoid is an oft repeated contention about scientists in general and mathematicians in particular but it is very difficult to prove or support on the facts.

Don't forget that life expectancy increased dramatically since the 1850s.
What was the life expectancy in America in say 1810?
So using an absolute age, such as 30 is suspect.
The figures would be drastically different if we considered age as a fraction of average life expectancy.

Secondly you need to allow for those, such as Galois in my example whose life was especially short. We do not know how his life would have panned out.
The patron of one of our newer members also suffered an abrupt early demise, before her mathematical work was done.

On the other hand Gauss, Euler and Maxwell were still going full belt on their death beds, at ripe old ages.

 

I know a long decline like a damped sinusoid is an oft repeated contention about scientists in general and mathematicians in particular but it is very difficult to prove or support on the facts.

Don't forget that life expectancy increased dramatically since the 1850s.

Has it? Particularly in regard to its relevance to this discussion?

In 1850 the average life expectancy at birth was 38.3 years while in 2000 is was 74.8 years (for white males in the U.S.). That sounds super dramatic -- a whopping 36.5 years nearly doubling the life expectancy. But that's life expectancy AT BIRTH and is heavily influenced by infant mortality rates. If you look at the life expectancy of people in those same years at the age of 10, it was 58 years and 75 years, respectively, and so more than half of that difference had been given back just by surviving early childhood. In fact, the major factor in the increase of overall life expectancy at birth has been increasing the chance of a child surviving to age 5. If you go over to the other end of the spectrum and look at the life expectance of someone that was 80 in each of those years, the difference is less than two years.

If you look throughout history at the age at which people die of "old age", the improvement hasn't been that dramatic at all. It was very common for adults to live into their 70s in ancient Greece and Rome both. Also, the numbers that I give above are misleading because they are for the population as a whole and do not take into account the differences between a farmer and a miner and a doctor and a mathematician. The life expectancies for people (adults) after you adjust for occupational hazards has been remarkably flat over the last several centuries. The gains have come less from medical science breakthroughs than from increases in workplace safety, primarily as a result of technology, but also because of workplace safety regulations.

What was the life expectancy in America in say 1810?

At birth it was quite low, somewhere around 35, but for someone that survived to age 40 it was 68 years compared to my life expectancy at age 40 which was 78. For someone that was 20 years old it was somewhere in the neighborhood of 65 years.

The figures would be drastically different if we considered age as a fraction of average life expectancy.

Probably not so different if you considered age as a fraction of life expectancy as of the age at which they did their work.

 

I know a long decline like a damped sinusoid is an oft repeated contention about scientists in general and mathematicians in particular but it is very difficult to prove or support on the facts.

Don't forget that life expectancy increased dramatically since the 1850s.
What was the life expectancy in America in say 1810?
So using an absolute age, such as 30 is suspect.
The figures would be drastically different if we considered age as a fraction of average life expectancy.

Secondly you need to allow for those, such as Galois in my example whose life was especially short. We do not know how his life would have panned out.
The patron of one of our newer members also suffered an abrupt early demise, before her mathematical work was done.

On the other hand Gauss, Euler and Maxwell were still going full belt on their death beds, at ripe old ages.

Gauss is one of my mathematical heroes... and I was just wondering about his work when I read your post... But there were actually two Guass out there, weren't they? Father and son, I think... haven't checked yet...

 

Borrowing a concept from Isaac Asimov…

The numbers 1 and 2 are well understood, but how about 1 over 2, or one half?

Can you hand me half a glass of water?

In theory, yes.

In practice, absolutely not, not without adding some additional definition (the glass holds 8 ounces and I gave you half of that or 4 ounces) constraint (I am within .1% of the nominal).

Still you end up with one glass of water with some arbitrary quantity of water within.

Where is the half?

I'm obviously missing the point. How does this indicate that 1/2 is any less understood than 1? If asked for a glass of water and the glass holds 8 ounces, then in practice the glass will actually hold some amount more than that and the glass I hand you will have 8 ounces within some constraint.

What about if I ask you for a PCB with 1 oz copper as opposed to 1/2 oz copper?

Or are you trying to parse things as "(half a glass) of water" as opposed to "half a (glass of water)"?

Even then, no two glasses are identical, so if you ask me to give you one glass and I randomly pick from a set of glasses, do you really know what you ended up with?

 

Why is that a failure? In fact, I'd think that if it weren't a never ending number, it would be as suspect. Take the set of real numbers. Remove all integers. You have an infinite number of numbers left. Repeat with numbers with one digit after the decimal point. An infinite set again. Repeat with n number of digits after the decimal point, where n = 2 to infinity. An infinite set. What's left? An infinite number of never - ending numbers. The fact that Pi and e, etc... are never ending comes as no surprise to me.

It may not come as a surprise, but would it really be suspect if it weren't a never ending number? Consider the ratio of the radius of a circuit to the length of the sides of an inscribed hexagon. That comes out not only to a rational number, but to an integer, namely 1. Is that suspect?

 

... Or are you trying to parse things as "(half a glass) of water" as opposed to "half a (glass of water)"?

And I had thought that LISP was a dead language...

 

And I had thought that LISP was a dead language...

As an interesting aside, I'm just getting ready to finish out the semester covering functional programming languages using Racket, which is a close descendent of LISP.