“There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy”
And of course Newton’s quote:
“I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”
We haven’t been at this for very long, on a galactic timescale.
Math can look very different and still give the same results. Algebra and geometry are “dual” - at bottom they look at the same problems from different aspects but the problems in one can be converted to problems in the other. (A very, very simplified definition.)
When string theory was new, several different looking variations were developed in 10-dimensional space until Edward Whitten realized they were all the same in 11-dimensional space.
Math is full of stuff like this. Fermat’s Last Theorem wasn’t proved for 350 years until bunches of new math that Fermat would never recognize were applied to it.
I’m sure that force would equal mass times acceleration no matter how the equation was written. The understanding came first and the equation later. But higher math may have been discovered and applied in various forms at various times to create math that we’ve never seen. Almost certainly set down in ways we wouldn’t have thought of. The Greek mathematicians didn’t write formulas as we know them today, but used words to describe their process. Formulas evolved slowly over centuries. Like life, evolution at that level is full of paths taken for no apparent reason while histories were dragged along because they couldn’t be removed even if less than desirable.
The actual way the world works would certainly be the same, but an alien’s understanding and description of it would certainly be very different. Others have mentioned the Periodic Table, for instance: If you look up the Periodic Table in the 1990-ish edition (maybe later, too: I haven’t checked) of the Encyclopedia Brittanica, you won’t recognize what you see. It’s the same fundamental information, of course, but presented very differently. And that’s with other humans, on our own Earth, with the same history of scientific discovery.
If we’re looking at “textbooks”, aliens might not even have “books” at all. Maybe their means of storing information is three-dimensional structures. Maybe they store information using lesser creatures that have been trained and/or bred to reproduce speech on command. Maybe their memories are so good that they don’t bother with inanimate information storage.
Math certainly wouldn’t be the same. Alien mathematics might not even have integers as a fundamental concept, for instance, instead treating entirely in real numbers. Why should they study integers? Until you get down to the quantum scale, there are no integers in nature.
This is the key point here. There is overwhelming evidence that fundamental physics – and therefore chemistry – is the same throughout the universe. There is zero evidence or even any credible hypothesis for how it could be fundamentally different.
This is emphatically not the same as trying to claim that “everywhere else in the universe, life must be just as it is here”. But it’s definitely saying that it must be based on the same physics.
Entirely created in the 20th century. But it raises a question relevant to the OP. Was its creation inevitable? It was created during the academic year 1939-40 when Saunders Mac Lane took a train from Boston where he was a junior fellow at Harvard to Ann Arbor Michigan to give a talk. In the audience was Samuel Eilenberg, newly arrived from Poland. Although Saunders and Sammy worked in very different fields (abstract algebra and algebraic topology, resp.) Sammy noticed that Saunders’ computations were identical to those on a problem he was working on. They got together to figure out how and why two such different areas of math could lead to the same computations, and thus created category theory. Even so, it lay dormant for around 15 years before taking off, largely as a result of a famous paper of Alexander Grothendieck.
While in many ways alien mathematics ought to resemble our own, it could look entirely different and have developed in a very different way. Easy to speculate, hard to tell.
Almost: theoretically it might be possible for different regions of the universe to work according to different physics. What doesn’t seem possible is for that to be true without it being obvious even across extra-galactic distances. Even small changes would have obvious effects. And according to some models there’d be a “domain wall” blocking our ability to see anything at all where the two universal regions meet.
It’s believed that if there are regions with different physics they exist well beyond the observable universe. Most likely due to cosmological inflation rapidly expanding the early universe so our observable universe was formed from an originally tiny region.
Their math might not be based on Base 10, it would depend on how many digits on how many hands they have. Base 2 might be the Universal math, though. They’ll certainly be familiar with gravity.
Would these be ‘atoms’ as we understand the term, though?
Isn’t there a calculation that above a certain atomic number, the electrons would need to be travelling faster than light…? I forget the details, I think it may have been Feynman?
I suppose that depends on how far you stretch your mind.
No strict integers in the sense of completely interchangable things, true. But groups of recognizably the ‘same type’ of object surely exist; which one would think almost certainly leads to an idea of ‘counting’?
It seems like you could infer them from the modes of a vibrating string well above the quantum level.
Yep.
Even if somehow integers didn’t arise early in their development of civilization & technology, they’d arise as soon as they invented set theory.
I could imagine an advanced enough mathematics that treated integers as a boring mostly forgotten subset of the reals. But I think they’d almost necessarily have been a foundation of very early arithmetic and hence organized / formalized math.
What kind of beings on what kind of world would need to exist to not recognize integers? Even if you assumed that the alien bodies were shapeless and had no arms, legs, heads, tails, or discrete internal organs and that they lived entirely in an unbroken ocean and had a hive mind so that the concept of an individual never arose, surely some feature in their environment would be countable and that would stand out for its weirdness.
If we’re going to have fun speculating: how about an intelligence based on potential gradients in a plasma cloud? (A bit like Hoyle’s Black Cloud, but with no solid bits).
It’s conceptually possible that math for them might be based on continuous fields rather than ‘objects’, perhaps?
And straying from math, for a moment: one thing which is almost certainly going to be different is colour (color in US). Maybe not the idea in the most general sense, but certainly in detail.
Our eyes are sensitive to a bit less than an octave of the electromagnetic spectrum. About 700nm to 380nm, if I recall correctly. And have three types of receptors for different ranges.
There are probably evolutionary reasons for this, but life forms elsewhere would almost certainly develop in a different way… if they have anything like eyes at all?
One might argue that the spectrum of typical stars falls in roughly this general range: so it could be likely that sensitivity in that area would be conducive to survival. But it would almost certainly be different in detail?
So how do plasma clouds see colors?
Expecting non-human intelligences to have the same mathematics as us is like expecting them to speak French. Mathematics is a tool to express mathematical concepts, but the details are entirely due to the incidental history. Translations should be expected.
Which is actually a quantum effect—the modes of a vibrating string exist because the boundary conditions quantize the energy states of the system.
But the fact that things can be countable doesn’t mean they necessarily will be counted.
It seems to me that math would be pretty much the same. Pi is still pi no
matter what number system you use. It might look different but it is still
pi. For example, pi in a base 8 number system is 3.1103755242₈ and
in binary it would be 11.00100100001111110110₂. but it is still pi.
(To be honest I used chatgpt to get these numbers).
Taking this seriously: perhaps they have sensors for a wide range of frequencies of electromagnetic radiation. And they might label or describe some ranges as ‘red’, ‘green’ or ‘blue’. In whatever communication language they use? Not the same ranges we use those terms for, of course…
Not all human civilizations have used based ten. We even have legacies from some of them - that’s why a circle has 360°, thanks ancient Babylon! They used a sexagesimal (base 60) system of counting, as did the Sumerians. The Celts used base 20, of which some linguistic remnants remain, such as the French “quatre-vingts” or literally “four score” for eighty. For that matter, using “score” in English for 20 might be a remnant of that - in Irish scór means “20”, and is pronounced the same as the English “score”. Base 20 was also found in Central American pre-Columbian exchange, such as among the Aztec and Maya. Arguably both the Ancient Central Americans and the Ancient Romans may have used based 5 before they used respectively base 20 and base 10.
I recall mentions of cultures using base 6 or base 27 (!?!) but haven’t been able to confirm the existence of those. It’s clear, though, that there’s nothing magic about base 10 even if yes, it is a common system used by people and the most common/popular these days.
So… a culture with six total digits on two “hands” that uses base 6 isn’t surprising, to use a recent cinematic example, but it shouldn’t be considered a requirement or the only possibility. Aliens might use base 12 (again - we have linguistic bits indicating that might have been something, with our dozens and a “gross”, that is a dozen dozen), or base 8, or 15, or… well, almost anything.
But the “base” of the math system is to a great extent a notational detail. The number twelve is still twelve whether it’s written as “twelve”, XII, C (hexadecimal), 20 (base six), 22 (base 5), gouges in a clay tablet by a scribe using base 60, or… you get the idea.
My understanding is that our sun produces larger amounts of light in the 400-700nm range compared to other ranges, but also our atmosphere and water allow 400-700nm light to pass through more easily and transparently. So for those reasons life here evolved to see in those ranges.
On a different sun with a different atmosphere or different water, then it may be a different range of light waves that are strongest and pass through the atmosphere and the water better.