Mathematics is both invented and discovered

I'm fascinated by the question of whether the laws of nature are "out there" in an objective external world, or "in here" within the subjective confines of the human brain.

A recent post on my other blog about male/female conversation styles mentions how I'd talk about this topic with another philosophically-minded man.

When men talk, most of the time they aren't trying to either reveal, or gain access to, inner feelings. My wife and I used to get another with another couple. The other guy and I would converse in one corner of our living room, while the wives huddled on the couch.

Our male conversation always focused on Grand Cosmic Subjects, like whether the laws of nature are actually "out there," or whether they're a manifestation of the human mind. We'd learn a lot about each other in this fashion.

Just not the same things the women would learn about.

After the other couple left, my wife would say something like, "How is Michael handling the death of his father?" I'd say, "He never mentioned anything about it. I didn't even know his father died."

This would astound Laurel. It seemed perfectly natural to me.

I bring up this personal vignette because it mirrors a long-standing debate among mathematicians: is their field of study invented or discovered? If invented, there's a subjective aspect to the laws of nature -- many of which, especially in physics, can be accurately modeled by mathematical equations.

This would be like my wife and her friend choosing to talk about one topic, while my friend and I choose to talk about a different topic. The content of our conversations would flow from what we wanted to focus on, not on an immutable Law of Living Room Discussions.

But many mathematicians are Platonists, believing that they are discove! ring num erical truths independent of the human mind. Here's how theoretical astrophysicist Mario Livio puts it in a fascinating Scientific American article, "Why Math Works."

At the core of this mystery [how math captures the natural world] lies an argument that mathematicians, physicists, philosophers, and cognitive scientists have had for centuries: Is math an invented set of tools, as Einstein believed? Or does it actually exist in some abstract realm, with humans merely discovering its truths?

Many great mathematicians -- including David Hilbert, Georg Cantor and the group known as Nicolas Bourbaki -- have shared Einstein's view, associated with a school of thought called Formalism. But other illustrious thinkers -- among them Godfrey Harold Hardy, Roger Penrose, and Kurt Godel -- have held the opposite view, Platonism.

So in a way the debate over the nature of mathematics is sort of like arguments about God. Is there really something we call "God" out there in some hidden sphere of existence, or is this notion a fabrication of the human mind?

Of course, there's a big difference between mathematics and God: the mathematical laws of nature accurately model many aspects of the natural world, while hypothesizing "God" doesn't add anything to our understanding of phenomena experienced in everyday life.

Still, I think those of us interested in the God-question can learn something from how Livio considers the issue of whether mathematics is invented or discovered. First I'll share his conclusion:

This debate about the nature of mathematics rages on today and seems to elude an answer. I believe that by asking simply whether mathematics is invented or discovered, we ignore the possibility of a more intricate answer: both invention and! discove ry play a crucial role.

I posit that together they account for why math works so well. Although eliminating the dichotomy between invention and discovery does not fully explain the unreasonable effectiveness of mathematics, the problem is so profound that even a partial step toward solving it is progress.

Let's be clear: as noted above I don't see any evidence that adding God or any other divinity into a description of non-human reality works at all, much less "so well." I just want to point out how even mathematics, the foundation of our most successful scientific discoveries, arguably has a subjective, invented component.

In that regard, I found this paragraph in Livio's article to be deeply thought-provoking.

Michael Atiyah, one of the greatest mathematicians of the 20th century, has presented an elegant thought experiment that reveals just how perception colors which mathematical concepts we embrace -- even ones as seemingly fundamental as numbers.

German mathematician Leopold Kronecker famously declared, "God created the natural numbers, all else is the work of man." But imagine if the intelligence in our world resided not with humankind but rather with a singular, isolated jellyfish, floating deep in the Pacific Ocean.

Everything in its experience would be continous, from the flow of the surrounding water to its fluctuating temperature and pressure. In such an environment, lacking individual objects or indeed anything discrete ,would the concept of number arise? If there were nothing to count, would numbers exist?

A few years ago I conducted my own thought experiment that went even further than reducing our world's consciousness to that of an isolated jellyfish. I said, Consider a cosmos with no consciousness.

When I pondered this, I had the same problem that popped up when I read about Kronecker's t! hought e xperiment. Mention of the jellyfish made me think, "Wow, what a stupid creature that would be. It wouldn't know what exists beyond the ocean. It wouldn't be able to perceive all the sights, sounds, smells, and such on land, in the air -- everything we humans experience."

But then I realized that my perspective was just that: mine. I was looking upon the jellyfish from my concious human point of view, even though the thought experiment was to envision all of the world's consciousness being embedded in a jellyfish.

In my blog post about a cosmos with no consciousness, I said:

Still, my thought experiment has led me somewhere, though not to my intended destination of an imagined cosmos with no consciousness. Ive understood that different sorts of consciousnesses are aware of the cosmos in different ways. We may not create reality, but our own unique perception of it is indeed created.

This is pretty much the same conclusion that Mario Livio arrived at in considering the question of why math works.

Mathematics is an intricate fusion of inventions and discoveries. Concepts are generally invented, and even though all the correct relations among them existed before their discovery, humans still chose which ones to study.

Ditto with God, though to a much greater degree.

Notions of God, supernaturalism, spirit, soul, enlightenment, salvation, heaven, and such are all conceptual. They're invented by people. There is no evidence, in contrast with mathematics, that seekers of the divine ever come into contact with anything but mental inventions.

So if even mathematics is a fusion of subjectivity and objectivity, religions are vastly more so. In fact, religiosity may be entirely invented by humans, with not a bit of discovery thrown into the dogma recipe.

I could be wrong about this. That's the beauty of a scientific outlook on life. Yo! u're always willing to say, "I could be wrong."