4 ms·

Things like this, or the advantages of Arabic numerals instead of Roman, make me wonder what sort of mathematical insights we're missing out on today due to our

by kej 10y ago

Things like this, or the advantages of Arabic numerals instead of Roman, make me wonder what sort of mathematical insights we're missing out on today due to our current notation.

mturmon 10y ago
By the way, this system of naming exponents looks a lot like Roman numerals to me. Pre-pending a name = raise to higher powers. It's a bit more ad hoc, because pre-pending a name multiplies the exponent (rather than adding, as Roman numerals do) -- making primes impossible to express. So they need the notion of a "sursolid" to express exponents (like 5 or 7) that do not factor into twos and threes. So, amazingly, he was able to create a system more ad hoc than Roman numerals.
noobermin 10y ago
Sure. Any choice might have some downside. One thing very interesting I always think about is, as I hinted at in my post, is the concept of series. Consider this example 1 + c_1 a^2 + c_2 a^3 + ... Common syntax makes multiplication between factors implicit, basically grouping factors together in a compact form. The exponents further compactify the visual presentation. This, in a way, makes each term in the series stand alone and appear as a unit. It seems then no coincidence that in physics and math we often consider partial sums as approximations, and concentrate on particular terms seperately. In physics for example, people think of "order \alpha^2 terms" or "higher order corrections" or physical quantities which are often taylor series cut off at some order, of course, assuming the quantity. A very familiar example of this are Feynman diagrams which are fancy taylor series in powers of force coupling constants. One wonders which came first. It may be that the notation followed this focus on individual terms, but it is interesting that no one considers "expanding" transition amplitudes in infinite products or continued fractions. Also, there certainly exists much more knowledge (theorems, technology) around series than products AFAIK. Again, I don't know which came first because I don't know too much about math history, but it seems reasonable that the causation may be reversed.
- thaumasiotes 10y ago
  > It seems then no coincidence that in physics and math we often consider partial sums as approximations, and concentrate on particular terms seperately. In physics for example, people think of "order \alpha^2 terms" or "higher order corrections" or physical quantities which are often taylor series cut off at some order, of course, assuming the quantity. There are very good reasons for this. The whole observation is basically the same phenomenon as someone observing, "you know, 34,825,119,276 and 35,174,884,395 are basically the same number for my purposes; I'll just call it 3e10 or, if I'm being really fancy, 3.5e10". In these applications, the series variable is a very small number. The higher exponents given to it in later terms of a taylor series cause those terms to be very small compared to the early terms. That's why we take the early terms as an approximation to the whole thing -- we have chosen our representation so that that will be true. (This is the entire reason for Taylor series in the first place -- a Taylor series is a Maclaurin series adjusted so that the variable can be small for purposes of the series, no matter what its absolute value might be.)
  - andrepd 10y ago
    Yes, I think the parent realises that. The point I think is that this stuff would be much harder to realise if we used verbose and prosaic descriptions instead of the visually suggestive modern notation (implicit multiplication etc)
    - thaumasiotes 10y ago
      As I read the parent comment, it suggests that the reason we think of the first few terms of a Taylor series as approximating the whole thing is that the notation suggests that the series is composed of a sequence of discrete terms. We know that this is wrong; Taylor series were developed so that the first few terms would approximate the whole.
- roywiggins 10y ago
  The notation came from trying to write down short polynomials, I think (quadratics, cubics, etc). Which before algebraic notation, was a huge pain in the butt. Series notation is just "this is a like a polynomial, but it goes on forever, so here are the first few terms."

by kej 10y ago

Things like this, or the advantages of Arabic numerals instead of Roman, make me wonder what sort of mathematical insights we're missing out on today due to our current notation.

mturmon 10y ago

By the way, this system of naming exponents looks a lot like Roman numerals to me. Pre-pending a name = raise to higher powers. It's a bit more ad hoc, because pre-pending a name multiplies the exponent (rather than adding, as Roman numerals do) -- making primes impossible to express. So they need the notion of a "sursolid" to express exponents (like 5 or 7) that do not factor into twos and threes. So, amazingly, he was able to create a system more ad hoc than Roman numerals.

noobermin 10y ago

Sure. Any choice might have some downside. One thing very interesting I always think about is, as I hinted at in my post, is the concept of series. Consider this example 1 + c_1 a^2 + c_2 a^3 + ... Common syntax makes multiplication between factors implicit, basically grouping factors together in a compact form. The exponents further compactify the visual presentation. This, in a way, makes each term in the series stand alone and appear as a unit. It seems then no coincidence that in physics and math we often consider partial sums as approximations, and concentrate on particular terms seperately. In physics for example, people think of "order \alpha^2 terms" or "higher order corrections" or physical quantities which are often taylor series cut off at some order, of course, assuming the quantity. A very familiar example of this are Feynman diagrams which are fancy taylor series in powers of force coupling constants. One wonders which came first. It may be that the notation followed this focus on individual terms, but it is interesting that no one considers "expanding" transition amplitudes in infinite products or continued fractions. Also, there certainly exists much more knowledge (theorems, technology) around series than products AFAIK. Again, I don't know which came first because I don't know too much about math history, but it seems reasonable that the causation may be reversed.

thaumasiotes 10y ago
> It seems then no coincidence that in physics and math we often consider partial sums as approximations, and concentrate on particular terms seperately. In physics for example, people think of "order \alpha^2 terms" or "higher order corrections" or physical quantities which are often taylor series cut off at some order, of course, assuming the quantity. There are very good reasons for this. The whole observation is basically the same phenomenon as someone observing, "you know, 34,825,119,276 and 35,174,884,395 are basically the same number for my purposes; I'll just call it 3e10 or, if I'm being really fancy, 3.5e10". In these applications, the series variable is a very small number. The higher exponents given to it in later terms of a taylor series cause those terms to be very small compared to the early terms. That's why we take the early terms as an approximation to the whole thing -- we have chosen our representation so that that will be true. (This is the entire reason for Taylor series in the first place -- a Taylor series is a Maclaurin series adjusted so that the variable can be small for purposes of the series, no matter what its absolute value might be.)
- andrepd 10y ago
  Yes, I think the parent realises that. The point I think is that this stuff would be much harder to realise if we used verbose and prosaic descriptions instead of the visually suggestive modern notation (implicit multiplication etc)
  - thaumasiotes 10y ago
    As I read the parent comment, it suggests that the reason we think of the first few terms of a Taylor series as approximating the whole thing is that the notation suggests that the series is composed of a sequence of discrete terms. We know that this is wrong; Taylor series were developed so that the first few terms would approximate the whole.
roywiggins 10y ago
The notation came from trying to write down short polynomials, I think (quadratics, cubics, etc). Which before algebraic notation, was a huge pain in the butt. Series notation is just "this is a like a polynomial, but it goes on forever, so here are the first few terms."