I've been waiting for this myself! Aren't these people educated?

I think I would listen to their names for the laws, and any stories they still remembered about why they had those names; and then tell them our names and the stories behind those. Most importantly I'd listen for any differences in emphasis. Consider for example the Peltier Effect, Seebeck Effect, and Thomson Effect. They're similar enough that they're often confused or just explained together; yet sometimes it's good to have the distinction. Alien versions of these effects would very likely have interesting shades of meaning, which would make it worth learning the aliens' own names.

For the most part though, I assume which names get used would be a matter of whose language you're speaking or who you're speaking to. Humans do tend to have instincts about what a thing is "really" called, which make it seem like more of a conundrum; but alien language intuitions might work much differently.

As for the credit for discovering -- for the most part, scientists recognize that there are more than one "real" discoverer of a thing. Names are attached because it makes them easy to refer to.

You second paragraph... you are seriously misunderstanding the nature of mathematics.

"7" is always seven. Always. Not the representation, the written numeral, the actual number of things. You count them, there are seven. It is always a prime number - it is not divisible by any other integer, no matter the representation or base used.

Being unaware of numbers... that would require some kind of hive mind or something, where individuals have no meaning at all, and even then, it still really makes no sense, as you need actual counts for all kinds of things if you are actually going to build something that works.

Topology, type theory, category theory, etc... those are *additional* areas of math, not replacements for the basics.

Unrelated: PLEASE fix your website. The captcha picture is chopped on the left hand side, so I have to make multiple attempts until I can get all the pictures in the rightmost few columns.

Sorry to hear that you have trouble posting, but I don't think I can do anything if it's the Disqus' issue. These guys don't even let me change the color scheme. I think the simplest solution would be to make a throwaway account so that the site won't ask you for captcha anymore.

When humans speak of founding mathematics on type theory, category theory etc., what we mean is considering that theory to be basic, and then building up the rest of mathematics from there. Typically Peano arithmetic would be the first thing to be built. The first time this happened historically was when Russel and Whitehead founded mathematics on set theory. (This foundation has more or less stuck; so in a technical sense set theory is more basic than numbers, and logic is more basic than set theory.)

Counting is basic to many *human* activities, and historically, numbers were the first big obsession of human mathematics (followed closely by geometry). I will admit, it seems to me counting would be pretty common among intelligent species. However, its familiarity and usefulness do not necessarily mean it would be universal. So I'll attempt in more detail to imagine why it might not occur to an intelligent species.

Counting and measurement are very top-down, control-oriented activities. If you want two things to be the same length, making a numerical measurement is actually a rather complicated way of doing it, which introduces error. Cutting them together or with a jig will do, and if you must measure, making a mark on a stick would work. Some of the earliest written numbers in human history were governments attempting to quantify farm productivity so that they could tax fairly. This was never really "fair" in the first place, since a variety of unquantified local conditions could affect what each farmer could afford to give up each year; the numbers were just appealing to human intuitions about fairness and serving as OK approximations.

I'm going to try and imagine a few precision processes, replacing numbers with tight feedback and non-numerical mathematical concepts. Bear in mind we humans shouldn't expect to always be able to imagine alien solutions to problems, so "I can't imagine doing X without numbers" is far from proof that an alien society would use numbers. In theory, we already know that complex mechanisms can be designed and built without using numbers, since DNA doesn't code for numbers anywhere. Anyway I'll do my best to give a plausible example.

A computer screen produces various colors by coding for intensity of red, green and blue. We humans use numbers for these, but numbers bounded to some space, for example from 0 and 255. An alien society less familiar with numbers might simply use various binary codes from (say) 00000000 to 11111111, conceptualizing them as lying on a line without thinking of them as numerical. The codes might coincide with our binary counting codes, or might be grey codes or some other system. When designing displays with more color depth, they would choose a code with more elements, but might conceptualize it as having colors between the colors of the older display, rather than giving a numerical color depth.

A human chemical factory relies on many numerical measurements to work. But picture for example a chamber which must be heated to a precise temperature. Imagine for a moment that a thermometer readout is used to regulate a heating element. All the numbers on the thermometer seem quite superfluous; what matters is that one line representing the correct temperature. Perhaps mass-manufactured thermometers would have lines labeling various important transitions, such as freezing and boiling points of common materials, and things like body temperature. (A medical thermometer might label danger zones above and below body temperature.)

Experts in mathematics or physics might be aware that there's an abstract sense in which different parts of the temperature scale can be compared; IE, a standard amount of energy could heat a material by what we would call "one degree" whether it's currently cold or hot. But this is a very abstract fact, which only holds true once you correct for changes in specific heat at different temperatures.

Even those experts might not believe the "degrees" would continue being meaningful forever (IE, they can count but believe there's some highest number). There are, in fact, human mathematicians who don't believe in infinity, and much of math works just fine for them. Essentially unjustified logical axioms are needed to prove there are infinitely many numbers.

It seems like much works out fine if numbers are just an obscure topic for certain experts. Admittedly such aliens would seem a bit dim-witted around certain topics, like perhaps estimating how long something will take. But humans are by default fairly bad at recognizing Lie algebras or finite simple groups in daily activities, and aliens who are raised on those concepts (but not numbers) might think humans are pretty daft.

As for the website issues, you might want to complain directly to Disqus somehow, since the problem is kind of on their end.

Sorry it took so long for me to notice there were messages in the spam folder.

"Those are just a few of the really crude-but-basic things serious
communication with an alien species requires. Mathematics is
foundational to all of them."

Yes, but none of those are the first and most important things, and the only things there tangential to the first and most important things (calorie requirements, and such) will matter before a group like this could figure them out, anyway.

My my, so much salt.
Did I said that nobody needs math? I said that I don't understand how math is supposed to help us to start learning structure of an alien language. How would you say what's your name in math?

It is, perhaps, good that you are not telling a story about serious
attempts at first contact and are doing a "spunky kids are better and smarter than the adults" one.

On the contrary, I've been trying to show how moronic Dan is at every opportunity. Without going too far, since I still need an alive protagonist. and yes, spoilers - all the big important talks will be carried out by other characters, not these four.

Aliens might think in what we would find a very unfamiliar logic without being aware of it. They might use some other structure than numbers for much of their math-like activities. Existing human mathematics might not help bridge the gap. Understanding them might eventually involve doing math, but it may be a new branch of math - and all "math" really means in that more general sense is that a task is difficult and requires writing down symbols.

You weren't supposed to see this. God damned post scheduling.

I've been learning omniscience at night.

Well done being impossibly early.