Ideas and constructs are real things. But perhaps not in the same way that physical objects or processes are. Many people believe that God is real in the same way that some believe “inifity” is real. Obviously as this board attests, there is some contention on this issue.

To assert that mathematics is “objective” is a different goal than to assert that the ideas expressed by mathematics are “real”. The objective part would require that any human sufficiently trainined in a type of math would draw the same conclusions, probably to a greater degree than natural languages seems to (English, Chinese, etc.), and to a much greater deal than beliefs, unfounded opinions, impressions or the much maligned intuition does.

To be considered objective, the test is more rigorous than these other schenarios. We might require that mathematical statements depict real phenomena, in order to be able to test whether they are objective.

In the above example of the n things in the universe, adding one more (this lat thing being non-existant) to arrive at n+1 begs the question: does this number depict anything “real” (existant). The answer is no, in terms of the number having any relevance. It does seem to exist as a concept, thus we have a dividing line between where objective math leaves off and speculative math begins. There are many such boundaries, perhaps they are infinite, although for concepts to be generated infinitely, we would need an infinite number of years, civilizations and brains (even non-human will do).

I’ll take the radical unfounded (expect by the piece mentioned above which I cannot track down) that infinity is a concept only, created by human math. Like the earth which approaches flatness at a sufficiently small scale, the universe seems infifnity when viewed on our scale. Not just an empty metaphor, a curved universe is a model that makes a finite universe possible AND allows for straight vectors to retrace themselves once they have gone far enough.

We humans are having a problem here modeling the universe on a grand scale. We need to imagine space like a large ball so that a finite universe still seems unbounded (a circle is unbounded but finite), or that space is linear in some way and has the property of infinity so that the damned end point with nothing beyond won’t be a problem.

If the sub-atomic is as conter-intuitive as we’ve found, no doubt the universe is too. The need for inifinty AND finiteness as concepts are grounded in human experience, and our own scale in the universe.