There is a classic question in philosophy of mathematics: Is mathematics invented or discovered?
But I suspect that question is only one instance of a much larger problem.
We obviously invent mathematical symbols.
The symbol “2” is something humans created. We could have used II, 二, two dots, or any other symbol. Mathematical notation is a language, and languages are constructed.
But the interesting question is not whether we invented the symbol.
It is whether we invented what the symbol refers to.
Did “two-ness” exist before any mind represented it?
If two planets orbited a star in a universe containing no conscious beings, would there still, in some meaningful sense, be two planets?
Or does “two” only come into existence once some mind develops the conceptual machinery required to recognize cardinality?
That question quickly becomes much stranger when we realize that mathematics is not special.
We can ask essentially the same question about almost everything we think exists.
I find it useful to distinguish three very different categories:
matter, concepts, and experiences.
And then ask the same question of each:
Can this exist independently of anything capable of experiencing or representing it?
1. Concepts
Mathematical objects are an unusually clean example.
There is little mystery about the existence of the symbol “2”. Somebody invented it.
But what about the thing it represents?
Perhaps mathematical objects exist independently of us and mathematicians discover them.
Perhaps they exist only as constructions within minds.
Perhaps neither description is quite right: what exists independently might be a space of possible structures, while minds invent particular languages for identifying and exploring them.
This becomes especially interesting once a formal system has been specified.
We may choose axioms, definitions and rules of inference, but we then encounter consequences we did not deliberately put there.
In that sense we appear to invent the entrance to a structure and then discover the landscape behind it.
But even that leaves the deeper question untouched:
Did that landscape exist before anybody entered it?
2. Matter
Here our intuitions usually reverse.
Most people have little difficulty imagining matter existing without observers.
The Earth existed before humans. Stars existed before life. A universe containing no conscious beings still seems perfectly conceivable.
That may well be correct.
But there is an epistemological peculiarity here that is easy to overlook.
We never experience matter independently of experience.
Everything we know empirically about the physical world arrives through perception, measurement, sensation and ultimately conscious experience.
We observe a screen.
We hear a sound.
We feel resistance when touching something.
We see an instrument displaying a number.
From these experiences we infer an external physical world that explains them.
That inference is extraordinarily successful. Modern science gives us powerful reasons to take physical models seriously.
But success should not be confused with direct access.
We cannot somehow leave experience, stand outside it, inspect objective reality itself and compare it with our representation.
There is always an interface.
So the proposition
“there is experience”
occupies a very different epistemic position from
“this experience is produced by a mind-independent material world possessing approximately the properties represented by our physical theories.”
The second may be overwhelmingly plausible.
But it is still an inference from the first.
And this creates a curious inversion.
We often regard matter as obviously real and consciousness as philosophically mysterious, even though consciousness is the one thing whose existence is presented directly to us.
Matter is what we infer.
Experience is what the inference occurs within.
3. Experience
And this is where, for me, the problem becomes most interesting.
The standard contemporary picture is roughly:
matter → brains → experiences
Physical processes occur. Some sufficiently complex physical systems become brains. Brains generate conscious experiences.
That may be exactly what happens.
But notice that this too is an ontological hypothesis.
We have extremely strong evidence that changes in brains correlate systematically with changes in experience.
Damage the visual cortex and vision changes.
Alter neurochemistry and subjective experience changes.
Stimulate particular neural systems and particular experiences can appear.
But correlation and dependence do not by themselves settle the deepest metaphysical question: what is the relationship between physical processes and experience?
Perhaps brains literally produce consciousness.
But other logical possibilities exist.
Perhaps brains instantiate conscious states.
Perhaps they constrain them.
Perhaps they select among possible experiences.
Perhaps matter and experience are two aspects of some deeper structure.
Or perhaps the relationship is even more radical than that: what we call physical objects might themselves be structures inferred from regularities within experience.
These possibilities are not equally supported by current evidence, and some are much harder to turn into testable theories than others.
But the important point is that the familiar causal story should not be confused with something we have observed directly.
Nobody has ever watched non-experience turn into experience.
We observe relationships between experiences, measurements and models of brains, and from them construct theories about what is happening.
Three kinds of reality
We therefore arrive at three apparently different domains:
Matter
Things that seem capable of existing independently of representation.
Concepts and structures
Things that may be represented by minds, but whose relations sometimes appear independent of the minds representing them.
Experiences
Things whose existence seems inseparable from there being something it is like for them to occur.
And there is no obvious reason to assume that these three categories have the same kind of existence.
Perhaps matter is fundamental and the other two emerge from it.
Perhaps experience is fundamental and our concept of matter emerges from regularities within experience.
Perhaps abstract structures have a kind of existence independent of both.
Perhaps all three are manifestations of something more primitive.
Or perhaps the very division into matter, concepts and experiences reflects limitations in the conceptual machinery with which we examine reality.
This is why I think the old question about whether mathematics is invented or discovered is much more interesting than it initially appears.
It forces us to distinguish symbols from what they symbolize, models from what they model, and representations from whatever — if anything — exists independently of representation.
Once we make that distinction, the question spreads everywhere.
Did numbers exist before mathematicians?
Did concepts exist before minds?
Did matter exist before experience?
Could experience exist without matter?
And perhaps the strangest question of all:
What would it even mean for something to exist if, in principle, nothing ever experienced it, represented it, interacted with it, or could distinguish a universe containing it from a universe without it?
Maybe numbers are not the strangest things whose independent existence we should be questioning.
Maybe the deeper question is not whether mathematics is invented or discovered.
It is what we mean when we say that anything exists at all.
