@axsvl77 Math is not related to our subitizing limit; if it were, then we'd use base-3 or 5 as that's where the limit is for most people. Math exist independently of it's user, it's a fundamental part of the universe just as physics is. So, regardless of their subitizing limit, they would still develop fundamental math, include arithmetics, it's unavoidable.

@axsvl77 Even if you have base-inf and the unique numbers Þ, Ð, and ®, then you still need to know their relationship if you want to use them. If you wish to tell someone to build ships so you can send an expedition of a total of ® individuals, but you can't put them all in one single ship, then you need to divide them in some way. So you say, build Ð ships with room for Þ each, because that totals to ®. You couldn't do that without arithmetics, regardless of how many unique numbers you have.

@axsvl77 You mention that arithmetics would be abstract and inconvenient, but all communication is abstract. Whatever you say, be it number or word, is simply an abstract representation of reality and not reality itself. We use abstract notation because it is convenient, even if it's tricky to learn and understand. It would be far more inconvenient to bring all individuals you wish to ship on your

expedition so that the engineers could see how many you need room for, when you can simply tell them "divide ® dudes into Ð ships"

@axsvl77 If I request 101 apples from a merchant, he wouldn't assume I used base-2 and only give me 9 apples; he'd know that I mean base-10 since that is our defined standard and I didn't state anything else (actually, chance is that he wouldn't even know that there are other base systems and just think that anyone who claims that 101 = 9 would be crazy). The same would be for an alien species using base-inf.

If they state that they want ž alien apples, then they wouldn't get 9 either for the same reasons (unless ž in base-inf happen to be 9 in base-10).

@axsvl77 But the base system is required in order to know what you mean. If we don't have a defined standard, regardless if it is base-2 or base-10 or base-inf, then you won't know what I mean if I say I need 101 of something as it depends on the base. So once a species develop math, it will eventually develop different base systems, but it would still keep it's defined standard when they communicate numbers naturally.

Math is a language and if you do not speak it, you cannot understand the world around you. All languages are abstract.

@Mrkvička Unfortunately it sounds like we are in agreement; your ships/dudes scenario is dead on. But what if this sort of arithmatic were inate to the thinking of the alien species, and did not require the rigors of well defined mathematics?

Certainly the mathematicians would define it arithmetic, but it would not be necessary to know this arithmetic to do the ships thing.

For a species that can subitize large numbers, how to split the people into groups would be obvious without thought.

Of course, this would not disturb the underlying mathematics.

@axsvl77 I am still going to be annoying and slightly disagree - unless there is an alien life form which is born with full knowledge of their language and science or have some other way of instant learning, then they to sit down and learn what each number is named and it's relation to other numbers, just as we learned it as young. I don't need to calculate that 27/9 = 3, I memorized it as I learned arithmetics so I do the split without thought.

But I still had to do the arithmetics at one point. So yes, the alien ship builders may not need to do active calculations when they build, but they will know about arithmetics and they have done the calculations at some point when they learned the name for each number and it's relation to all other numbers.

@Mrkvička We are very limited creatures with our only ~100,000 word vocabulary and small subitization. We are even more limited with only 80 years of life. Perhaps the theoretical creature lives long enough that spending a few decades just to learn number names isn't a big deal? And even more, why cant an alien life form be born knowing their language?

There is absolutely no reason to assign our faults to others.