In a solar system in a totally different galaxy, would their books on math, physics and chemistry be identical to ours

[Exapno_Mapcase]

We can look at the history of math from various cultures on this planet, before the invention of formulas.

The earliest math problems were word puzzle, like this one from the Babylonians about 1850 BCE:

I found a stone but did not weigh it. After I added one-seventh of its weight, And one-eleventh of this new weight, The total was one ma-na. What was the original weight?

The Babylonians tried to convert all fractions so that they had one in the numerator, which they then referred to pre-calculated tables. A very different mindset.

Later mathematicians also used word puzzles but in forms more easily translatable into the modern. The Persian mathematician al-Khwarizmi* wrote al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala, or The Compendious Book on Calculation by Completion and Balancing, in the early 9th century. It’s chock full of word puzzles like this one:

If a person puts such a question to you as: “I have divided ten into two parts, and multiplying one of these by the other, the result was twenty-one;” then you know that one of the two parts is thing, and the other ten minus thing. Multiply, therefore, thing by ten minus thing; then you have ten things minus a square, which is equal to twenty-one.

Dan Rockmore explains:

“Thing” more familiarly goes by the label x, which came into use centuries later, as did other symbolic shorthands like + and = and the superscript “2” to denote a square. In modern algebraic language, what al-Khwarizmi tells us is that x(10–x) = 21, but also x(10–x) = 10x–x² (what you get when you multiply “thing” by “ten minus thing”), so that 10x–x² = 21.

We can easily see that two answers are needed and that they are 3 and 7.

Though earthian trends led to universal formulas, the pathway could have gone to unfamiliar forms. But guessing about what the most advanced unrecognizable math might be is an unnecessary complication, like guessing about a keystone’s size and position before the arch is designed.

* several bad transliterations later, that begat “algorithm”.

[Hari_Seldon]

@xtenkfarpl

And of course if we want to drill really deep: Godel destroyed Hilbert’s dream that all mathematics could be reduced to logic and symbol manipulation.

That’s not correct. All of mathematics is logic and symbol manipulation in a formal sense. What Gödel did was show that if every statement in an axiom system broad enough to include ordinary arithmetic, could be resolved (shown true or false) in that system, then those axioms were inconsistent: you could prove that 0 = 1.

Maybe it is just a failure of my imagination, but it is inconceivable to me that you could base mathematics on the reals and not have the notion of counting. You could not have polynomials for example, since they have degrees. Or analytic functions since they are integer-indexed power series. Or polygons.

[Der_Trihs]

@Mama_Zappa

Years back, I read a SF story (by H Beam Piper, I think) in which the ruins of a long-dead civilization on Mars were discovered.

The “Rosetta Stone” was something like a periodic table.

Yes, the short story Omnilingual. (checks) mentioned upthread in fact.

It’s at the higher levels that chemistry would vary on other worlds, both because of the environment and because it’s an extremely broad field. There’s an immense number or ways to put atoms together, and an immense number of ways to do so; it seems likely that entire fields of chemistry popular on Earth could be ignored in favor of ones we’ve never bothered to get into.

[LSLGuy]

Good point.

Human’s so called “organic chemistry” starts getting seriously complicated about the time it slides into “biochemistry”. The molecules that make Earth life go are much larger, more complex, and more varied than simple high school level discussions of e.g. CuSO₄ and FeO₂ can imagine. Heck even rather simple sucrose is C₁₂H₂₂O₁₁ and systematically described as “β-D-Fructofuranosyl α-D-glucopyranoside”.

Given aliens of a similar or higher science / technology level but very different biology, we should expect them to have similarly gone deep into the details of what they’d call “biochemistry” while knowing bupkiss about ours.

And vice versa.

[xtenkfarpl]

@LSLGuy

Given aliens of a similar or higher science / technology level but very different biology, we should expect them to have similarly gone deep into the details of what they’d call “biochemistry” while knowing bupkiss about ours.

Almost certainly… though it does seem that some molecules that are at least percursors of things like amino acids have been detected in meteorites and perhaps by spectroscopy in interstellar clouds.

Maybe these just happen to be rather likely probabilities of carbon chemistry?

[xtenkfarpl]

@Hari_Seldon

That’s not correct. All of mathematics is logic and symbol manipulation in a formal sense.

True. I was oversimplifying. But it seems to me that the Hilbert (and Russel-Whitehead) idea was that all mathematics could be in principle reduced to a formal generative system? Even if this is impossible to realize in practice?

[Der_Trihs]

@xtenkfarpl

Almost certainly… though it does seem that some molecules that are at least percursors of things like amino acids have been detected in meteorites and perhaps by spectroscopy in interstellar clouds.

Maybe these just happen to be rather likely probabilities of carbon chemistry?

Yes; many of the “building blocks of life” are substances that are just naturally likely to occur if conditions allow it, which is likely why they became the building blocks of life in the first place.

[Hari_Seldon]

All of mathematics can be reduced to formal logical principles from axioms. But most are not decidable. One that is decidable is the language of formal inference, using primitives like p ==> q. Every statement can be verified or refuted using a truth table.

I was thinking about the possibility of a mathematics with real numbers, but not whole numbers. It really does make sense. If you are studying, say functions from f: R → R, you will surely want to talk about the derivative f’. But then f’ may have a derivative f’‘, and a third f’‘’. What do you call them? Second and third derivative don’t work if you don’t have a concept of two or three. Then you will want Taylor series, which are indexed by the natural numbers. I just do not see how this could all work, except at the most superficial level.

[Exapno_Mapcase]

@Hari_Seldon

Maybe it is just a failure of my imagination, but it is inconceivable to me that you could base mathematics on the reals and not have the notion of counting. You could not have polynomials for example, since they have degrees. Or analytic functions since they are integer-indexed power series. Or polygons.

Thank you for this and later posts. That’s the point I thought I made repeatedly, but it’s better to hear it from a real mathematician.

[Der_Trihs]

Going the psychological route, it occurs to me that a species that naturally had what we would call a “savant” style ability to perform mathematics would likely treat mathematics different in books. A lot of what we consider basic mathematics wouldn’t be taught by books (outside of the terminology) any more than we teach things like the exact muscles to flex in order to dance out of a book. It would be a “learn by doing” matter that they had little conscious awareness of.

Eventually they’d likely formalize such math and write it down, but even then it would be relegated to classifications like abstract mathematical theory, computing or neurology. Not “here’s what to teach kids”.

[xtenkfarpl]

@Hari_Seldon

If you are studying, say functions from f: R → R, you will surely want to talk about the derivative f’. But then f’ may have a derivative f’‘, and a third f’‘’. What do you call them? Second and third derivative don’t work if you don’t have a concept of two or three.

Isn’t there a branch of math called fractional calculus where you can have something like a ‘halfth’ derivative? Would it make sense to say that first and second derivatives are just special cases of that range? Disclaimer: I am not a real mathematician…

[Melbourne]

@Hari_Seldon

If you are studying, say functions from f: R → R, you will surely want to talk about the derivative f’. But then f’ may have a derivative f’‘, and a third f’‘’. What do you call them? Second and third derivative don’t work if you don’t have a concept of two or three.

You call them grand deriviaive, and great grand derivative, and great great grand derivative. Or Rat, Ox, Tiger, Rabbit, Dragon, Snake, Horse, Goat, Monkey, Rooster, Dog and Pig.

The shift from ‘Roman’ numerals to ‘Arabic’ numerals made a lot of arithmetic easier, and certainly numbers are critical to the way we understand derivatives, but they aren’t necessary. A lot of math can be done with labels, or geometry, and some math is actually more interesting when detached from numerals.

[Hari_Seldon]

I am unaware of any such branch, but who knows what someone might decide to examine.

[xtenkfarpl]

I originally heard of it from a YouTube video, I think. Not the most reliable source, obviously!

But Wikipedia has an article on it:

Fractional calculus

[Chronos]

Fractional derivatives aren’t actually that difficult. You can turn any function into its Fourier transform, and you can take the derivative of a sine function just by translating it horizontally by the right amount. So, Fourier transform, translate a fractional amount, and transform back.

You can also get real-number mathematics by starting with the first way humans tried to formalize mathematics, through geometry. A number, then, is just a line segment of a particular length, and of course that length can be any real number. You can develop that into a notion of integers, as in fact Euclid did, but it’s really awkward, and he probably wouldn’t have if there wasn’t already a less-formalized notion of integers floating around, that he wanted to describe.

[xtenkfarpl]

@Chronos

Fractional derivatives aren’t actually that difficult. You can turn any function into its Fourier transform, and you can take the derivative of a sine function just by translating it horizontally by the right amount. So, Fourier transform, translate a fractional amount, and transform back.

I think it works more easily using Laplace transforms, though? But we’re getting rather off-topic now: maybe time for a different thread.

As I’ve said: I’m not a professional pure mathematician. I’m an engineer with a good grounding in math up to undergrad level. But higher math is a wonderful spectator sport - as Fred Pohl said about science in general…

[Hari_Seldon]

The Wiki article is interesting; it was new to me. I still cannot imagine anyone finding those formulas who was unaware of the ordinary derivatives and integrals. Note that the Cauchy integral formula, based on ordinary integration, was used in the definition.

[Mikestone8]

This point has been made in Science Fiction,

In “Omnilingual” by H Beam Piper, astronauts on Mars are baffled by te Martian “alphabet” until they happen upon a Periodic Table, which is of course the same as ours.

[hajario]

@Chronos

No. Like I mentioned, the periodic table in the Encyclopedia Brittanica would be unrecognizable to most Americans.

Still curious what you meant by this

[LSLGuy]

Third cite is the charm. See also post #10 & 83.