Quote:
Originally Posted by Robh
Perhaps, it is not clear exactly what my argument (opinion) is and maybe we agree at some level.
I believe that mathematical axioms, conjectures and theorems exist as a conceptual logical structure independent of the physical make-up of our Universe. Yes, you need a Universe and a conscious being to develop the logical structure but it exists as an independent body of ideas.
Initially, the concept of number probably originates from the need to count objects. Once defined, number can exist by itself e.g. 1<2 and 2<3 then 1<3 does not depend on any physical object but simply on the definition of "<". Mathematics can be applied to the Universe but is not discoverable in the sense that one discovers a physical law or theory (e.g. General Relativity), which through observations and measurement, we deem to govern some aspect of the Universe.
However, as I mentioned earlier I also think that the language of mathematics is communicable across the Universe (or, hypothetically, over to another Universe) to another intelligent being as it does not depend on the physical properties of that Universe. In this sense, the logic of mathematics is invariant of the Universe's material structure.
But I do not agree that there is an embedded wallpaper or backdrop of mathematical knowledge waiting to be discovered, like gold in a field. Mathematics is a conceptual body of logical theorems that are driven by human interest. The results attained will vary according to the path followed and they are not simply waiting there to be discovered. A particular path followed does not necessarily have to be relevant or have the need to be discovered in another Universe, even though the concepts may be communicable. Take, for example, the whole body of theory around complex numbers. This was developed as an interesting concept based on a definition of i (i^2=-1) and later on found applications, but was it a discovery or simply an invention? Another being in another Universe could live without it; it wasn't lying around waiting to be found.
Consider this. Someone invents the idea of a perfect number- a number that is the sum of all its factors other than itself. Example: 6 has the factors 1,2,3 and 6=1+2+3; 28 has the factors 1,2,4,7,14 and 28=1+2+4+7+14. This is a defined and constructed concept; it wasn't waiting around to be found. Any subsequent logical derivations or proofs exist from the initial definition. The human mind is capable of thinking of and defining whole new directions- from perfect numbers we can navigate to amicable numbers.
So, although number may be a consequence of the Universe at large, most mathematical discoveries (conjectures and theorems) will depend on a particular path followed starting from defined statements and axioms . These are not embedded in the Universe waiting to be "discovered" but are driven by human interest. Hypothetically, a comparison of the mathematics of two cultures from different Universes may find some overlap of mathematical theorems but I would tend to think they would differ significantly. However, the basic logical starting points from number would allow each culture to navigate the others mathematical directions and results.
At some level the logic is transferable but the content is not universal.
Regards, Rob.
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Rob,
Consider geometrical theorems/conjectures. The catalyst for geometrical concepts is through measurement. In a subtle way a geometrical theorem is analogous to a scientific theory except that individual measuresurements themselves do not prove the theorem.
As an example consider the Pythagorean theorem for right angle triangles c^2=a^2+b^2.
Did Pythagoras invent this geometric property for right angle triangles? Clearly no.
Mathematicians before Pythagoras knew of the relationship on the basis of measurement but where not able to prove it for all "a" and "b". Before Pythagoras the geometrical property was based on conjecture.
The geometrical property has always been around, measurement has simply confirmed it's existence.
The element of invention is through proof. There are at least 8 ways to prove the Pythagorean theorem. Each method is a product of logical processes at work but the ultimate objective of each proof is to confirm the existence of the mathematical property.
This example can be extended to mathematics in general.
In essence the theorems are "already there", how we proceed to prove them is where invention comes into the picture.
Regards
Steven