Godel’s incompleteness theorems show that every mathematical system has theorems that can’t be proven in the system. Dark Matter and Energy present a similar challenge to the standard model of physics. Even if mathematicians can slather a system theory of explanation for dark matter and energy onto the problem there is no necessity that it be true. It may be phenomenal and coincidentally, as might it’s verification. Science or knowledge so awaits observation and pragmatic reckoning plus metaphysics ideas.
Math systems are incomplete and that math systems and physics differ substantial hence mathematical logic has a shaky, conditional relationship with physical systems it is used to help build a formalized model of. The unknown is an inherent part. Of the Universe and applying incomplete systems to construct models of it should generate more Incompleteness and uncertainty.
Godel’d Incompleteness paradigm;
It is what it is. Formal mathematical systems need to beg the premise of a few unproven axioms to be in business. Those items can be incorrect and if applied to the real world invalidate complete systems. Riemannian geometry for example might work better than Euclidian for select applications in Hilbert space.
Errors in math at any stage can invalidate answers just as wrong premises in logic provide wrong conclusions. Mathematical models are always just that…nothing more than analogies of what is assumed about physical cosmology. When Newton or Einstein get the premises wrong on how things work so do the mathematicians that relied on the physicists ideas to construct models. Pure math usually can’t discover anything besides more math. As glorious as math is sometimes it is an accomplice to discovery that can go way wrong if it’s axioms were wrong or the physical cosmology misunderstood in the beginning.
I am not of the opinion that math is useless for science. Cantor’s trans-finite numbers discoveries for example, were a tremendous, enlightening advancement for theoretical knowledge that has probably not been applied to the ‘real’ world to a fraction of what the theorem may.
It is possible that trans-finite, scalar infinities could be quantified for quantum field applications…who can say? Celebrate infinites instead of hating them. I think it plain though math applications to the real world require knowledge of the real world. Discovery tends to be a physical event, and mathematics haven’t yet been credited with founding reality.
Of course some mathematicians virtually believe math is the foundation of reality- Math is a language something like others including computer languages. Those fluent may wish to exagerate their importance.
Math is a language that most don’t speak fluently. So uninterpreted properly and applied in a meaningful way to real world issues, events etc including cosmology math is as useless as speaking in tongues is for a church congregation when it is not properly interpreted. Without looking through the (microwave) telescope not even the c.m.b would be known.
Galileo also saw a few interesting things with optics. Without Tycho Brahe’s observation records would Copernicus have been credited for proving heleocentricity?
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Garrison Gibson: books and blog 