We obviously cannot be sure of anything alien intelligence might or might not do. Nevertheless, the subject here is math. A mathematics without countable numbers is as alien as we can imagine. Simple dimensions, foundational to virtually all math we use, require counting. Foundational to physics as well, since 2D physics is quite different from 3D physics. An intelligent alien with advanced mathematics that couldn’t tell the difference would, to say the least, have a difficult life.
Yes, counting is foundational to the way our species handles dimensionality. I’m not convinced it’s a necessary feature of all possible conceptualizations of dimensionality. I think we’re fish swimming in the water of our mathematics, which makes it very difficult to imagine approaches outside of it.
But why? Take sheep, for example: Do you actually care what integer number of sheep you have? Or do you care about the amount of mutton and wool? Both of those will be real numbers, or at the least things that can be better approximated by real numbers than by integers.
OK, true. So the alien real-users might still eventually come up with the notion of integers. But it might still be an esoteric, specialized concept for them, with their “ordinary typical numbers” being reals.
Sure, if they use pi. But even many human mathematicians argue that we shouldn’t, because tau (twice pi) is the more fundamental quantity.
Isn’t twice anything making use of an integer?
No. Twice a real is still a real.
For this discussion, there’s values like 2 that are just another real number in a continuous set. And there’s values of 2 as a member of a countable, discrete set.
Heck the “eyes” we use to see the universe outside our own solar system (telescopes, spectrometers, cameras, etc) see colors entirely differently than our eyes would if we were close enough. Those colorful images of the Crab Nebula look nothing like what we would see with our own eyes if we were close enough.
These are the only ones of which the news has come to Harvard
And there may be many others but they haven’t been discovered
Eh, if you ask some random people, you will be hard-pressed even to find one who can define what are “real numbers”.
Tom’s song needs a new verse these days, it seems… 
ISTM we’ve named a lot of elements we haven’t created yet. Easy enough to declare that atomic number 345 is henceforth LSLGuyium. Rather harder to whip up a batch.
That’s absolutely true, but the answer isn’t just that they often color-shift astronomy photos (including false-color images from frequencies above or below the limits of human perception).
The other difference is that nebulas are very, very dim. And getting closer doesn’t help with this. The effect of getting closer is exactly cancelled out by the fact the it takes up a wider field of view.
If we ever had a Star Trek type spaceship that could fly around the galaxy, the view would look much like it does here on Earth. Even if you flew right up to a nebula, you wouldn’t notice anything with the naked eye.
For example, the Andomeda Galaxy as viewed from the Earth is actually much larger than the full moon (roughly six times wider). Despite the fact that it is about 2.5 million light years away. But also very, very dim.
Here’s a photo comparing the relative sizes as viewed from Earth. Andomeda is greatly brightened, but not magnified relative to the moon in this photo.
With the naked eye, you can only see the bright galactic core, which looks like a smeared-out star. You cannot see the galactic arms, because they are too dim. And counterintuitively, you couldn’t see them any better if you got closer.
Bottom line, the only way to get a good look at a nebula or a galaxy is to let a camera collect light from it for a good long while. Or use a telescope, which also collects more light than your eye can. But even with a telescope, you have to also connect it to a camera with a long exposure to see the fantastic images you see in the media of nebulas and galaxies.
You’re using earthian math terms. We don’t know if the aliens recognize a difference between reals and intergers and subsume the latter.
@Chronos used “twice” in his post when referring to tau. That shows how foundational integers are our mathematical thinking. Twice implies, in fact is literally derived from, two times. Two times implies three times, four times, n times. “Times” implies discrete integers in a countable set.
Moreover, the reals include not just integers, but the rationals, which emerged, in our thinking, from division into parts. Even if the reals emerged first in their system, a back-formation into rationals and integers would seem to be basic extrapolation for any mathematical system.
System is the important word. That their math and physics (counting protons, neutrons, and electrons, e.g.) and chemistry (counting molecular bonds, e.g.) and therefore all sciences give identical results to ours is axiomatic in this thread. Therefore, their system must include similar concepts even if expressed differently.
Handwaving away the most basic parts of their system is allowable for an unknown, but once you throw parts out for no reason how can any discussion proceed about what the system is like, which is the point of the thread.
Yes, but to be fair that’s just a variation of base 10. 1-9 was essentially a character repeated as many times as the number it represented. 10 was the next unique character. 11 was represented by the character for 10 plus the character for 1. 20 was two 10 characters. 36, for example, was three 10 characters plus six 1 characters.
A bit arbitrary, really. The ‘base’ used for arithmetic isn’t really a fundemental property of arithmetic, it’s just a particular notational convention. For practical purposes it doesn’t really matter at all.
Which is why all that ‘new math’ confusion with diffent base drills was a waste of time. For everyday use, it doesn’t matter: the idea that everyone has to understand the deeper underlying principles was… misguided. And usually rather silly: it often boiled down to just adding the phrase “the set of” to existing material.
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.
I can imagine an intelligent species that has a brain that can organically calculate any math problem instinctively. They would have no need to record those functions in whatever their approximation of a book is or even have an underlying understanding of how those conclusions are made.
An interesting question. If such things were actually produced in supernovae, and are long term stable, might there not be a miniscule amount of them on Earth (like rare heavy elements)?
The question then is: would we notice them? Perhaps we don’t have detectors that might spot them… or we haven’t developed such instruments yet?
In a way I can see that, Just as we don’t have to calculate differential equations in order to catch a thrown ball. What kind of civilization or society (if those concepts even make sense in that context) might arise from that, I wonder?
Except that I’m pretty sure that notational convention evolved before complex math. We use base 10 because we have 10 fingers
Marvin_the_Martian opined that it might matter ‘on how many extremities they have’. To which Broomstick offered the base 60 alternative. I was just pointing out that it’s essentially a base 10, and I’m pretty sure it’s because they had 10 fingers.
It’s surmised that base 20 was used because that culture counted toes as well. Some cultures used the thumb to count the 12 individual joints (phalanges) on the other four fingers of one hand to get base 12. Some groups, like the Yuki of California, used the spaces between fingers rather than the fingers themselves to count - base 8.
So most early counting systems relied on digits. I didn’t contend that a base is fundamental property of math, only that what ever that base is is likely to have grown organically. I think it’s probable that alien math would have grown from a base organized around their biology.
No matter how advanced alien life form may be now - they were once ‘cave man’ equivalents.
Possibly not. In 5-6 thousand years of recorded human civilization, our own concepts of these systems has changed a lot–chemistry and physics more than math–and the way we view them in the next few hundred years could be different yet.
Consider the colors of the rainbow. We collectively agree that these are red, orange, yellow, green, blue, violet, and indigo. But these are arbitrary choices of parts of a gradual and shifting spectrum. There could be isolated tribes that see ochre, teal, and taupe as the primary colors and who’s to say they are wrong? Different species perceive different color ranges.
Any system for categorizing anything is based on some arbitrary choices. We differentiate between mammals and birds in a way the ancients probably didn’t, and aliens may differentiate between elements and fundamental forces along different lines than we do.
Exactly. I think we’re agreeing here. Integers can be an extrapolation of the reals. An extrapolation means it’s not necessary.
Mathematical system can have the values of 2, 4, 17, etc without having a distinct concept of integers. That lack may hinder the development of some science and technology.
How much did calculus impact our science? But our species certainly had science and technology before we had calculus. Similarly, an alien science may have stalled until they figured out that non-continuous sets are extremely useful. But they’d still have science and technology before that point.