Stanford has a Metaphysical Lab in which an algorithm generates "abstract objects". I have commented that Leibnitz's mathematical infinitesimals may be considered a kind of metaphysical actuality, but others disagree. There are now rigorous logical structures embodying them.

A major obstacle here is there is no clear definition of metaphysical. Some might say that String theory is metaphysics, others might disagree. Mathematics beyond infinitesimals, or parts of mathematics could be considered metaphysics. I consider transfinite set theory and all its offshoots to be a kind of metaphysics, and there are oodles of proofs of theorems therein.

However, I am handicapped by having been a mathematician and not having been a philosopher, so take my comments with a salty grain. :cool:

Could it be possible to translate metaphysical concepts into matematical language so that we would be able to prove some theories as valid and refute other? — Eremit

I certainly find that any good metaphysics has a mathematical clarity. So there is a deep connection. But not a simple one.

In general, good metaphysics provides a putative model of reality in terms of some self-consistent logical argument. So it is mathematical in the sense of having some logic at work. And that logic can be checked out in terms of the mechanics of how it maps the inputs of an argument to its outputs. There is a system of rules. And so an argument can be checked for validity or internal consistency under those rules.

So - as with maths - you can prove the validity of the argument. But then if metaphysics wants to be saying something fundamentally true of reality - ontology being its ultimate goal - it has to start engaging with science and empiricism.

The wrinkle is that maths can now be regarded as itself the science of patterns. It is actual research into the nature of being - if Being itself is constrained by a generalised demand of self-consistency.

That is why fundamental physics and the maths of symmetry are in such tight alignment. Metaphysically, both treat their "realities" - the worlds they seek to explain - as "systems". And symmetries are the key that unlocks that door.

So that is an example of how science, maths and metaphysics become joined at the hip once they seek out a particular line of self-consistent logic - a holism that makes maths "unreasonably effective" for the physicist.

There are lots of domains of maths that have very little metaphysical application. They lack the kind of logical (hence causal) closure that metaphysicians seek in making their ontological models of reality.

And it is not generally helpful to think it is just a case of translating verbal descriptions into mathematical language. Metaphysics is generally going to be in advance of some mathematical notation. The mathematical concepts have to come first. And when first understood, they may still be hazy. But good metaphysics always has that quality of someone describing an essentially mathematical structure. There is some holistic arrangement of parts that forms a holistic process. And the solidity of that discovered structure - as with a symmetry operation - is something that can then be turned into a standardised mathematical notation.

Do you think that we could describe metaphysical theories by geometric constructions? — Eremit

I am arguing that all good metaphysics is basically about the holism of structures. So that is geometric in that you would have to "picture" both the relata and the relations.

But every good geometric idea can be describe algebraically (and vice versa). So it is not that geometry is primary. The key again is the presumption that we are trying to discover the hidden structure of reality - its deep skeleton. And the maths that is about the science of patterns, the logic of structures, is naturally going to be treading the same path.

It will be making concrete the useful mathematical language that can then be employed by mathematical physicists to write their fundamental theories. And the fundamental physicists are today's actual metaphysical community.

The academics working in philosophy departments can contribute plenty to a history of metaphysical ideas, but not much to the development of new ideas beyond some useful commentary from the sidelines.

In summary, metaphysics, maths and science are in practice fused at the cutting edge of human inquiry already. Mathematical strength thought has always stood behind the best metaphysics. And science created the empirical connection between such metaphysical models and the reality they might claim to model. A lot of bad metaphysics fell by the wayside as a result. And the mathematical scientists were left the ones closest to the new action.

Some might say that String theory is metaphysics, others might disagree. — jgill

This is a good example of the tensions now developing because mathematical research looks to be cutting deeper than empirical research.

We simply can't recreate the Big Bang to test out our theories. But we can computer simulate the Big Bang - or at least check the self-consistency of a mathematical model.

So the tensions are about the trivial thing of academic careers. Are your skills obsolete if Cern has no new super-collider, but Stanford has this new supercomputer simulation team? You will be quick to start name-calling - "That's not real science! That's just metaphysics!" - if your own job is on the line.

Academia is a social game. People have to construct their in-groups by "othering". Calling someone a metaphysician can seem like the worst kind of insult.

It is not helped by the fact that many then take advantage of the social notion that metaphysics means "unbounded speculation" rather than logically rigorous argumentation. Any crackpot will claim to be a metaphysician as convenient cover.

But it is all games. No need to worry. The simple fact is that the traditions of metaphysics, maths and science have always been fused at the cutting edge of human inquiry. There are few examples of great scientists who didn't combine the three in productive fashion.

And the fundamental physicists are today's actual metaphysical community. — apokrisis

That's what I would have guessed.

What if mathematics IS metaphysics? — Eremit

Could not be, ever since Riemann and others discovered the logical consistency of non-Euclidean geometry in the 1840's. Prior to then it was believed that mathematics was physics and that the world conformed to the Euclidean paradigm. Riemann discovered that other geometries were possible; and then Einstein showed that not only were they possible, but that they were actually true.

Mathematics can not say what the truth of the universe is. Math can only say which mathematical ideas are logically consistent. And there are logically consistent but mutually inconsistent theories out there.

Even in contemporary physics there are many interesting new theories out there that all use math but can't all be true. Math is just a tool for modeling ideas. It can't tell you what's true.

:up:

And there are logically consistent but mutually inconsistent theories out there. — fishfry

But Euclidean geometry was merely shown to be a special limit case on non-Euclidean geometry.

The two were consistent. The advance was to find a parameter - a constraint on parallel lines - that could be relaxed to arrive at an even more symmetric or generalised mathematical structure.

And in that, the maths rather exactly mirrored the physics.

The everyday world looks flat and Euclidean - at the inertial scale we are likely to be measuring it. That only proves to be the special case as we become able to step back and factor in gravitational and quantum "curvature". The flatness becomes relative to the dynamics of spacetime and energy density.

So the abstraction in terms of mathematics - the relaxation of some critical constraint on the model - mirrors the actual thermal evolution of the hot Big Bang. Go "further back" towards the Planck scale and relativistic and quantum effects intrude. The geometry loses its cold and expanded Euclidean flatness. It becomes the chaos of quantum gravity - equal parts black hole strength curvature and quantum strength uncertainty.

Mathematics can not say what the truth of the universe is. — fishfry

What the steps from Newton to Einstein and so on tells us is that the Ancient Greeks were already on the right track with Euclid. His geometric model of space was the correct starting point. It was simply over-specified in not having time and energy explicitly included in the dynamics.

Once the symmetries were expanded to include these - once space could be bendy, and then bend as a precise reciprocal of its energy density - we had stumbled into the model that could describe a GR world.

Quantum field theory adds another dimension of plasticity to the frozen Euclidean realm. Instead of being all bendy, now the spacetime manifold is all "grainy" - composed of fluctuations that need to be collapsed.

QG is already clear as the final step that would unite GR and QFT in a still higher state of geometric symmetry. Bendy space would be unified with grainy space as two sides of the one dynamical coin.

The great metaphysical project has been working out really well. Ancient Greeks got the ball rolling. Modern mathematical science has now got the hang of its deep logic.

All aboard for the ride. :up:

Mathematics can not say what the truth of the universe is — fishfry

And that presumably is what metaphysics is all about. Or is it? :chin:

And there are logically consistent but mutually inconsistent theories out there. — fishfry

Right. It's the same thing with religions, for example (closer to my home). There are many religious systems consistent within temselves, but inconsistent with other systems. And then we ask which one is true, if any?

Why couldn't they all be true? But to answer that we would have to define the truth. Are we talking about some absolute truth or relative? Is there such thing as absolute truth?

Metaphysics is generally going to be in advance of some mathematical notation — apokrisis

But not entirely, as seen in applications of non-Euclidean geometry. Perhaps there is a "mathematical universe" behind the physical one we see and a mathematician's subconscious makes a connection and forms a pathway for those ideas to enter our consciousness (along with a lot of garbage, of course!), and this in turn sparks metaphysical flames.

The difference between the two is that the metaphysical ideas can not be proved whatsoever. — Eremit

Well, let's see, that's not really true. Being proved whatsoever obviously needs clarification. Using a truncated definition of Metaphysics ( the nature of existence or existing things) consider a couple quick thoughts.

In consciousness, you have metaphysical phenomena. Universal things like the experience of love, the will to survive, self-awareness, wonder, etc. that we can't really explain in a concrete way, yet they exist in our minds. Are they abstract feelings or experiences? In part they are, because they represent a sensory response which in-turn words cannot really capture or describe, in their entirety. The true nature of the sensory experience transcends common logic. We just use abstract language to convey their meaning.

The other thought relates to engineering formulas and/or gravitational formulas. Are those abstract, why sure they are. Do they represent concrete things? Mathematical engineering formulas can, like in the world of structural engineering. Structural beams are described/designed initially using math. Maybe not so much in gravitational forces/physics itself, but the similarities to abstract languages(s) still apply.

I think where you would have difficulties the most, is where you try to abstractly describe the nature of conscious existence and self-awareness, using mathematical abstracts. For example, viz the human condition, if the commonly referred EM fields of consciousness can capture the nature of the so-called universal phenomenon of love, then there would not be a need for abstract thinking. In other words, what purpose does abstract thinking serve; why does it exist?

In consciousness, you have metaphysical phenomena. Universal things like the experience of love, the will to survive, self-awareness, wonder, etc — 3017amen

So are sensations and emotions part of the "truth" of the universe? Are they metaphysical, like String theory with its mathematical context? I remain curious about modern set theory. Has that been inspired and gifted to mankind in preparation for a future revelation about a truth of the universe that will indeed embody what seems now to be only faintly related to our physical world? :chin:

I would be curious to see if any of the topics discussed here as metaphysical would arise as "abstract objects" from the algorithm at the Stanford Metaphysical Lab. Anybody here from Stanford?

So are sensations and emotions part of the "truth" of the universe? Are they metaphysical, like String theory with its mathematical context? — jgill

Very nice!

Mmmm, well 'truth' is a loaded question as they say, but a very good question nonetheless! Revelation starts with asking the right questions, I think. Kind of like Wheeler's game of 20-questions and PAP :chin:

Perhaps the existential truth of the universe is simply conscious existence (self-aware beings). Consciousness, mostly, is a phenomena that includes those metaphysical features of our awareness. And those are without question abstract existing things. And so, in large part, mathematics is also another metaphysical thing or language because they both seem very abstract and non-material, as mentioned previously. They are things that seem logically necessary for existence, yet abstract and not concrete nor do they confer any real biological advantages.

String theory, (combining gravity with QM if I remember correctly) i think, would have to take into consideration conscious existence because its purpose is to calculate a ToE. And because its means/method of using mathematical abstracts is similar to consciousness, it seems like the proper methodology to use. I hope it can be successful some day, but am not too optimistic that we mere mortals can achieve such heights of explaining a type of 'cosmological consciousness' as it were. (I hope I'm wrong, in which case 'true' revelation may occur... .)

But what if... What if mathematics IS metaphysics? — Eremit

I think that Metaphysics is all about Abstractions in the human mind, and Mathematics is about as abstract as you can get. So yes, "mathematics is metaphysics". But I am not qualified to prove that metaphorical similie mathematically. Perhaps the Metaphysical Lab mentioned by jgill will shed some light on the subject. Since modern philosophy is mostly concerned with Metaphysics (e.g. Consciousness), some mathematical evidence may bring it closer to the fold of empirical science. :smile: