Philosophical / Scientific Question regarding reality

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I would agree with you that mathematics could describe none-existent universes or other none-existent things but the mathematics itself is real even if it has no physical counterpart.

A mathematician could analyse the mathematics and reduce it to first principles and verify the mathematics is true.

I'm making a distinction between pure mathematics and the application of it to describe physical phenomenon.

There is no such thing as an apple. Our senses convey to us individual reactions to stimuli and we relate these stimuli to create a class of objects which is an abstraction. Each object of what we call an apple is a unique object separate and unique from all other objects we call an apple. There is no such thing as two apples in reality. There are two objects which we fit into the abstract class "apple" but even a gross examination of these objects will reveal huge differences. But the human mind cannot live in and deal with a world where every object is an individual object. So we, instead, invent a world of objects many of which are the same and this fictional world is the world of mathematics and the world of our "reality".

We each live in a world comprised of the inputs to our sense apparatus. We automatically associate these inputs to objects and create classes of similarities amongst objects. These classes are abstracts and not real objects. There is no such thing as two apples. Even a gross examination of two unique objects we call apples will reveal huge differences and some similarities. But in mathematics we deal only with the similarities and call the objects two "apples" but two apples don't exist in reality because each apple is a unique object. But the human mind cannot make sense of an infinite number of unique objects so we simplify the world into classes. This is Plato's world and it is a mere construction of convenience because we cannot operate in any other world.

Sorry, there was a computer glitch and the two posts are almost the same.

I agree fully with your post. I didn't know it was Plato's world though - my knowledge of classical Western philosophers is limited. One thing though, your use of mathematics here is in the applied sense. Throughout this thread I'm thinking only of pure mathematics. Mathematics not applied to anything at all.

So in pure mathematics 1 + 1 = 2, similarly Pythagoras's theorem in the pure sense is a fact independent of the human who discovered it. The same theorem would exist if discovered by an alien race or not discovered by any intelligent life forms. It is pure logic. My argument in this thread is that pure mathematics and pure logic have an existence independent of the universe and observer. The fact that mathematics gets discovered by humans and it is used to describe physical phenomena is irrelevant to my argument. The fact that the mathematics is used "badly" to count superficially similar objects is also irrelevant to my point.

I'm still not sure about the typewriters and monkeys and all the World's literature. Mathematics can be discovered by anyone - or it could even be discovered by an alien race. Mathematics is universal, it isn't a discrete phenomenon.

It is something of an assumption that all minds work similarly. Even here on Earth there are species that sense the world so differently that their modes of thinking would surely develop an abstraction language quite different from that humans use. When creatures use sound to see with the world is quite different and the way they group their classes would probably be different. A fly lives at a different time rate than a human. That a fly mathematician might use similar methods for building abstract models of the world might indicate that there is an optimum process for doing so but that this optimum process constitutes some sort of existence independent of the universe seems to me a rather strange concept of existence. In a different universe with different fundamental constants (assuming life there is possible) mathematics might be quite different. Einstein's use of non-Euclidean mathematics to bring Newton up to date indicates that certain processes of working with abstract models can radically change the paradigms of reality.

Incidentally, a mathematics where 1+1=10 is rather common today.

Your are reasoning without a brain?

(... paragraph with silly jokes ... cut ...)?

Our brains are the result of the reality surrounding us. The same logic govering mathematics govers the interaction of proteins in our brians. It is therefore not surprising that the model we developed fits to this logic. So maths has physical basis in the structure of our brains.

When we say that ideas, spoken less poetic, are physical states of brains, a) would stand.

It is only "pure logic" if you apply basic axioms. In an other system of axioms a x b is unequal b x a (which just the case if a and b are vectors). Our mathematics is a construct, not a discovery. We use this construct to describe nature.

Well, that's debatable. It may be a function of axioms, but the point is that e.g. the Pythagorean Theorem holds irrespective. Anyone who assumes the axioms of Euclidean geometry has to accept the Pythagorean Theorem.

The Pythagorean Theorem is a special case of the Euclidean metric. Although many mathematical statements deal with two dimensional space, they are readily generalized to three or more dimensional ones.


Right, but that's not important. Real arithmetic in binary is precisely the same as that in decimal, differing only in how we label the terms. Math has moved quite beyond labels; that two structures which we may perceive to be different can be isomorphic to one another is a basic concept in modern algebra. Math doesn't care about labels; it cares about structures and patterns.


Well, the fact is that no formalization of math could possibly have the final verdict on every conceivable mathematical statement. That is, an axiomitization of math will not be able to express consistently the truth value of every mathematical proposition. This is sometimes used as a rationale for a Platonic interpretation of math; that because what we conceive of as math is not reducible to axioms and logic, it cannot be viewed as a trivial consequence of human-constructed assumptions; hence math is not equivalent to a language.

Different sense systems perceive different patterns and different relationships between patterns. That there is a similarity between many of these relationships does not endow an existence outside of the minds that perceive these patterns. Our universe works within a set of observable rules. Different universes probably have other rules. Whether there is a set of rules that applies to all universes seems to me speculative.

But, eg., there could be no universe in which it is possible to make a formal system sufficiently powerful to model arithmetic which was both complete and consistent. All universes will obey the unconditional truths of math out of necessity. Although how a particular universe may behave is unknown, or what kinds of mathematical systems some animal may think of are unfathomable, to a certain extent what is possible is dictated by things that can be discovered a priori. It would be most odd if mathematics could tell us how the universe could look yet it is not simultaneously "real".

Whatever the hell "real" is.

Well that's really the problem with this whole debate. Once we define real, the debate becomes trivial.

My point is that mathematics in book or verbal or other symbolic form is a language that becomes "real" in the mind of someone who interprets the symbols. It means nothing to a horse or a dragonfly or someone not involved with mathematics. It's a method of communication of the inter-relationship of patterns that has an internal consistency that is rigid. Nothing more.

Real "in here" (i.e. in the Mind) is not necessarily real "out there" (in Nature independent of sentient beings).

Mathematics is an artifact freely invented by humans. If it happens to help us describe nature, that is a happenstance. It just so happens to be the case.

ruveyn

But math isn't the language, it's the referent. The two can and must be separated as a consequence of the Incompleteness Theorems, because what is conceived of as mathematics cannot be reduced to trivial relationships among assumptions. Mathematical propositions are more than syntax.