Quantum Mathematics?

Question

As per my last question, this has less to do with cold, hard, and fast calculations and more to do with the interplay between mathematics and philosophy...but as armchair philosophers aren't as hard to come by as knowledgeable mathematicians, I'm asking this here.

Schrodinger's thought experiment about the cat came to the conclusion that the cat in the box was both alive and dead at the same time until it was actually observed, and that it was the observation of the cat that made reality collapse into only one of those outcomes (either the cat is alive, or the cat is dead, but not both.)

What I take away from that is that simply our observations can have cause/effect relationships to phenomena at the quantum level. Whether my understanding of this concept is a little too simplistic or not, I'll leave up to the opinions of those who are more knowledgeable in this matter than I.

At any rate, if such is the case, then what I'm pondering is this:

Let's assume that accurate mathematical models can be made of the inner workings of subatomic particles and their interactions. If simply by observing those particles we can change how they act, would that not mean that observation itself affects certain areas of mathematics? That certain relationships and concepts in math change based on our observance of them?

And again, just like my last question, I'm not looking for reinforcement here; I know my conclusions have to be incorrect, and I'd be interested to find out why from a mathematical point of view that's more finely honed than my own. Thanks!

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Ah, I get what you guys are saying. Mathematics exists as a descriptor of objects/ideas/concepts, and nothing else...even when what it is describing is itself.

So while observation would affect objects, and perhaps to some extent ideas, observation would do absolutely nothing to the actual description of those objects and ideas, which is why mathematics wouldn't be affected.

I knew was going wrong somewhere - thanks for drawing it out, friends!

Answer 1

Suppose you look at a cow and say "That cow is eating grass.' But the cow hears you, and looks up startled! It's now not eating grass.

Does any of this change what you mean by "eating grass"?

Answer 2

Even if you assume we have accurate mathematical models to describe the physics of particle interactions, remember that mathematics is merely serving as a language to "model" these systems. Moreover, the act of observation should be worked into the model, as observation is a physical process (usually observing something requires shooting photons at it). In particular, the mathematical formalism of quantum mechanics basically works observers into the picture via self-adjoint operators on the Hilbert space of states of the system you are considering. On the other hand I don't know what it means to "observe mathematics", or how mathematics could be affected by observation. Remember that physics always requires making some sort of real-world measurement, while mathematics never requires making any such measurements.

Answer 3

Mathematics is not a physical thing. It is based on reason, not measurement. For example, if I tell you that I have $\$5.00$ and you know you have $\$10.00$ you can deduce that we have $\$15.00$ without making a single measurement.

It is true that our daily experience guides our intuition. Mathematics captures that. Consider the case of parallel lines. We added the parallel postulate to geometry because our intuition of what happens in a plane demanded it. Even if we could do some kind of measurement in our physical world that proved that plane geometry was a figment of our imagination, everything we know about plane geometry would still be true.

Mathematics does not say what the world is made of, it is just a language for describing it.