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A professor once told me, for example, that the act of counting is widely regarded as a first sign of mathematical intelligence. Is it though?, can the act of counting (or performing simple arithmetic operations such as division or multiplication) be thought of as a sign of mathematical intelligence among a group of species?

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There is no humans who don't have math intelligence. Prof is wrong. Because HIGHEST mathematical intelligence is language and imagination. Mathematics is a small(lets call it half) part of language because without language it is a car without driver. –  Asphir Dom Apr 15 at 10:58
    
@AsphirDom We were not only talking about humans, but rather animals (or even plants) and other beings. –  Miguelgondu Apr 15 at 17:49
    
@AsphirDom: whether or not the highest mathematical intelligence is language and imagination has no obvious bearing on what the first sign of mathematical intelligence might be. Did you think perhaps that the claim was something to do with the status of mathematical intelligence itself, among skills that humans have? –  Niel de Beaudrap Apr 17 at 9:24

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Can it be? Of course! Counting falls under the sphere of skills and knowledge that we typically call "mathematics". A number of animals have rudimentary skills of this sort (even pigeons; this is at least estimation-of-number if not counting-one-two-three, which if accurate falls under mathematics (cardinality of small sets)). You can even implement it with biochemistry (hence plants); you might not want it call that "intelligence" though just because it's not implemented with neurons but with cellular processes.

Do we want to use the term mathematical intelligence this way? That's a more complex and subjective question. If you're interested in highly reflective and axomatized mathematics (e.g. ZFC), probably not. In that case you probably don't want to call it mathematical intelligence when people count, add, etc., either; abstract axomatic mathematics is rather different, in terms of the type of thinking required, than formulaic skills like counting or long division.

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I think one should be careful not to conflate an ability to judge numerosity with the ability to count, the former being the ability to give the number of things without counting. Also, I don't think the neuron/other cell distinction is the right one, every physical system can be thought of as "implementing" some maths, even if that physical system is a collection of neurons the cases where one would consider them to be counting in is limited to only a few (i.e. when they are embodied in such a way as to be able to respond to physical objects - however they are defined) –  Lucas Apr 14 at 19:56
    
@Lucas - Counting small numbers is not conceptually distinct from numerosity of small numbers, is it? Also, although physical systems can be described with math, only a small fraction of them are usefully described as an abstract representation of some external stimulus, and since "intelligence" normally refers to cognitive processes one ought at least require neurons to be doing it, no? –  Rex Kerr Apr 14 at 21:16
    
Hmmm, difficult. The cognitive process of counting and the process of judging numerosity are clearly quite different. A large-pinch-of-salt analogy would be as algorithms where counting is O(n) and judgments of numerosity are O(1) - one "just knows" how many things there are. I think most humans have integrated the two things quite well, we can just see four things and treat them like we can were we to have counted them - in fact, we generally don't count small numbers of things. As the two are interchangable for us it is tempting to equate them, this is potentially a mistake in animals. –  Lucas Apr 14 at 21:53
    
@Lucas - Fair enough. But judging numerosity is also a mathematical task. –  Rex Kerr Apr 14 at 21:58
    
In the case of more things/less things, like the article you referenced I'd say that's not clear. If they were judging brightness, and therefore more or less photons, perhaps even ordering them, would that constitute a mathematical task? I think it can be reduced to absurdity. Personally, I'd need to see a degree of rule based generalisation to call it maths - I'm fairly sure one could get pigeons to demonstrate this to a degree that would satisfy me, but judgments of numerosity will retain the "perceptual" label. –  Lucas Apr 14 at 22:15

Counting is probably the most basic mathematical operation; and need not involve arithmetic; but probably should involve order.

One should be careful in distinguishing this from recognition of similarity & differance.

After all, a hawk can distinguish a rabbit from a pigeon - they are different, and recognise that one pigeon is not much different from another.

Likewise, it may recognise that two pigeons and similar to another two pigeons, and is different from six pigeons.

But it isn't counting until it can place them in order - and one can presume that is so, if the hawk recognises that six pigeons is more than two pigeons.

I can't resist adding, though it is irrelevant to the intent of the question, that counting doesn't stop there, after the invention of set theory, Cantor showed that you can count infinite sets, and place them too in order. So even a basic idea like counting can be revolutionised after the many millenia since it was first invented.

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