Our entire system of mathematics is founded on the concept of unity, and the basic premise that one plus one equals two. Take fractions, for instance. 'one', or 'unity', is subdivided into OTHER 'unities' which are no longer unities, but sub-unities? Why are they not just broken down into more 'unities', such that '1 broken up = 2', or '1 becomes 5', or '1 = 58' in some operation '=' called 'breaking' or 'becomes'? 'Unity' is indeed a very abstract concept that other minds might not develop the same way the human mind has.
Our concept of 'unity' is based entirely on 'intuition' and 'supposition'. A system of math based on an entirely different supposition would be entirely different. We take it for granted, but it was not always so. For instance, the entire concept of counting would be different. More like Roman Numerals, where every 'quantity' would have a different symbol, unrelated to the previous symbol, and there would be no sequential progression from one to the other. In point of fact, some posit that Roman Numerals evolved the symbol 'V' to represent 'a hand's worth' (five fingers) and 'X' 'both hands worth' (ten fingers). Would a person born with six fingers on the hand instead of five see 'V' and 'X' as different amounts than someone with five fingers? If the idea was to use a 'one-to-one' correlation between fingers and the objects being 'counted', one can easily imagine two people, one with five fingers and one with six fingers on their hands, could be in dispute over what 'one hand's worth' would be, and how to represent it in symbols.
Had we never developed the Arabic number system, our concept of mathematics would be very different indeed. We now take it for granted, but it is NOT that old a concept, as mathematics goes.
European Expansion The first mention of Arabic numbers in the West is
found in the "Codex Vigilanus," a historical account of Hispania
published in 976. Pope Sylvester II began to spread knowledge of
Arabic numerals throughout Europe beginning in the 980s. As a student,
Sylvester studied a form of mathematics and requested that Italian and
Algerian scholars translate some of the earlier mathematical texts
into common European languages. This was accomplished more fully in
1202 with a book by Leonardo of Pisa called "Liber Abaci."
The turning point in our mathematical system, and what made it the way it is, was the invention of the concept of 'zero'. Making the absence of anything an actual number, and assigning it a value, changed everything. That came as a natural progression from 1+1=2 to 1-1= well, what?
Another turning point, which developed only because our minds were able to handle extreme abstract concepts, was algebra. The concept or representing the 'unknown' by a symbol. There have been studies that relate how the development of algebraic concepts is dependent on the unique properties of a subset of human minds and human cognition. The fact that there are many, many humans that just can not comprehend algebra at all indicates exactly how much it is dependent on constructs and cognitive abilities that have evolved in human minds.
Why did the development of algebra lag behind geometry for so many
centuries? Why do today’s pupils have difficulties with even the
simplest word problems? What prevented generations of mathematicians
from accepting the idea of the irrational and the negative numbers?
What are the roots of the difficulties experienced by students
confronted with the concept of complex number for the first time?
It is neither by chance, nor by mere carelessness, that my list of
questions is a mixture of psychological and historical puzzles. As
different as they seem at first glance, these two sets of problems may
in fact have much in common. Indeed, there are good reasons to expect
that, when scrutinized, the phylogeny and ontogeny of mathematics will
reveal more than marginal similarities. At least, this is what follows
from the constructivist view according to which learning consists in
the reconstruction of knowledge.
Consider also out mathematical symbol "=". Exactly what does it mean? For instance, it has an entirely different meaning in computer science ('becomes') than in algebra ('has the same value as') and in expressions ('replace with' or 'substitute') and in topology ('transforms into'). A mind that is based on patterns and wholistic processing instead of sequential processes and logical step-wise processing would have a very different concept of 'equivalence', with a very different symbolic representation.
It can not be absolutely expected that every intelligent mind would make progressions in mathematical thought in the same way that we have.
