Geometry deals a lot about mathmatics of our dimensions of space. But the laws of logic don't require dimensions at all. There could be a reality where there weren't any dimensions at all so they couldn't have geometry (although if there were no dimensions, there would be no life so nothing even could think of it if it was there). So if it were possible for them to not have geometry, could you conclude that geometry is just part of our laws of physics?
2It is not a law of logic. It is the mathematical theory of space, that can be applied to physical space.– Mauro ALLEGRANZANov 2, 2022 at 1:09
2I cannot think of anyone who thought that geometry was part of logic, even the logicist Frege did not. Geometry has the same status as analytic dynamics. One can take them as abstract mathematical theories, and some parts of them are only that, or one can take them as mathematical models of physical phenomena, in which case their laws are laws of physics. The geometry of Minkowsky space is a model of special relativity, for example, and the structure of its cones then expresses the law that the speed of light is the limiting speed.– ConifoldNov 2, 2022 at 2:30
Yes, absolutely. That is Mathematical Anti-realism (ie, that math doesn't have a seperate realness, like say Mathematical Platonists believe). I disuss how numberlines can be thought of as abstractions of continuous symmetry operations, here: Could our natural numbers be non-standard?
But, mathematics is not just numbers. Logic is clearly important. The view that mathematics reduces to logic, is called Logicism. Personally, I see logic as a domain of abstraction, one that specifically considers the entailment of systems that can be unambiguously modelled, in a way that makes alternative histories where another of a set of options or decisions would have happened explicitly, or been very likely. I think of the ruthless logic of electrical circuits, or computer code. Coders I know react very strongly against the idea what they do is math. It's about if/then/or, and comparing the entailment of explicit rigorous models to results in the world (and so logical deduction). Discussed here: Why is a measured true value “TRUE”?
The 'laws of logic' are to me, suspect. It's a legacy of axiomatic thinking. Look at the Parallel Postulate. It seemed like a 'self evident elementary truth', then you get Minkowski space, and the Relativistic recognition that Euclidean space is not the default. The Law of the Excluded Middle is similarly suspect. 'Natural numbers' are about what is intuitive, not what is fundamental, discussed here Relationship between real quantities and numbers, and similarly with a given set of axioms or rules - as Wittgenstein inveighed, look to the game, to understand the rules. I'd look to Hofstadter's work, to Strange Loops, and the idea of Tangled Hierarchies, to see that, more interesting than law-based axiomatic thinking, are feedback and emergence, what happens beyond First Order Logic. As I see it we aim towards a meaning-cosmology that situates us, so as to be able to act effectively. We try to interface between our sections of understanding, with more and more versatile concepts. See
- Is the idea that "Everything is energy" even coherent?
- Is the idea of a causal chain physical (or even scientific)?
- Why is the universe governed by very few laws of high generality instead of lots of particular ones?
Another way I would caution about overconfidence in regard to the scope of geometry, is to consider entropy. I would argue modern science is property-dualist, between mass-energy, and entropy. The assumption has been entropy is a secondary characteristic of mass-energy, simply it's arrangements. But as Quantum-gravity has stayed stubbornly out of reach, it seems increasingly clear information flow is a key bridging framework: propagation of measurements in QFT, light-speed signals in Relativity. Wheeler proposed the It From But paradigm. Loop Quantum Gravity works with what we see as emergent from a spin-lattice network, time and space as emergent from a shifting flow of information. And my favourite proposed direction Constructor Theory, looks at what physical transformations of systems are possible as fundamental, the topology of their symmetry operations. So, instead of entropy getting subsumed into geometry, we may find time as emergent from entropy, as discussed here: Are the concept of time and space apriori to natural language or are they just references within natural language? with I think the implication space dimensions are emergent too, following Noether's Theorem that they can be understood as the result of sets of continuous symmetries equivalent to conservation laws. Modern physics requires we seperate time as a dimension from time-ordering, because of the relativity of simultaneity. We have to understand logic, and the entailment of events, seperately from the idea of reference to a universal clock, in the same way Minkowski space leaves behind neatly ordered Euclidean space; and the Bell Inequalities change how we understand locality.
For me the 'universality' of geometry is really about what experiences are intersubjective. We can use different forms of logic based on different experiences and intuitions, but we all necessarily experience consequences of specific degrees-of-freedom - if we didn't, our chemistry wouldn't work. Intersubjectivity is the real key to understanding a 'global' or shared reality with laws in common between experiences, it is the symmetry operation of 'if I were you and you were me'. The co-arising of mind and world, of self and other. Logic and mathematics are sets of abstractions, the domain of the Private Language Argument, and so necessarily they exist in this intersubjective space where we are trying to play certain language games, to extract transferable information, and find processing shortcuts like local or continuous symmetries, to 'chunk' our experiences and make our communication more capable.
See Do we create knowledge? and According to the major theories of concepts, where do meanings come from?
Geometry is derived from mathematics i.e., a particular form of geometry can be associated with a particular form of algebra within mathematics. In this sense geometry has nothing to do with logic, but lots to do with the parts of physics which have algebras embedded in their description.
Mathematics is infinite -- the number of possible postulates, and resulting derived mathematics principles, is uncountable.
Geometry is one category of derived consequences. There are multiple possible geometries. That one, or several, of them manifest in our world, IS "physics", or at least an empirical discovery. It is not logic.
What mathematics, if any, and what LOGIC (logic is also infinite, like math is) manifests in our world is a CONTINGENT fact, it is not a logically derivable one.
Geometry is the art (technique) of measuring the land (geo-metry).
This is somewhat lost on English speakers, but is apparent in French, for example, where "géomètres" means land surveyors.
By extension, geometry is the logic of space, and by further, relativist, extension, the logic of space-time.
The purely formal treatment of the logic of space or space-time is geometry in the mathematical sense.
Thus, mathematical geometry is derived from the technique of the land surveyors of Euclid's time.
We had to wait for Einstein, a scientist, not a mathematician, to arrive at a good mathematical model of the geometry of space-time.