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Mathematics major vs mathematics degree?

Regarding math vs physics, concept vs reality, the legitimacy of the foundations of mathematics?

  • I am a mathematicians and quite well versed in it. This is not some fifth grade childs question that cant or wont wrap his mind around math. I hope I get serious respondents from people who can think deeply enough about my question. More of an observation and discussion topic than a question. Most of modern mathematics, including calculus and number theory and so on... rests on Euclidean geometry in one way or another. The notion of flat space. Not only flat space, but limited to two or three dimensions as well. All of math stems from these premises or axioms or however you want to view them. Even the value of pi isnt just defined as the ratio between circumference and diameter, but specifically in two-space. But then here comes along Einstein who proved that space is not flat, it is curved. And m-theory which proposes up to 11 spacial dimensions. This is reality. But why arent the foundations of mathematics, Euclidean geometry, questioned and even tossed out? How can we reason about our world, apply math to science, when the math is fundamentally inconsistent with the reality it is used to reason about? At what point do we or should we abandon mathematics? In logic, the law of non-contradiction is a fundamental rule. And yet, real observations of the quantum physics, which forms the building blocks of all this universe, demonstrate self-contradictory behavior. The number line is said to be continuous. Continuity is a significant notion in mathematics. We can always break down mathematical reality into smaller and smaller intervals. But this is not true of physical reality. The planck constant, planck length, so on, is the *actual* smallest. The universe is not continuous, it is granular and it discrete. And there is the dilemma.

  • Answer:

    Mathematics is not reality. Physics uses mathematics to model reality, and over time, the fit is pretty good. But it is, and probably always will be, an approximation. Newtonian physics was an approximation to reality. Einstein improved the approximation; but Einstein's physics is also an approximation.Reality does not contradict mathematics, it contradicts the physics used to model it. We can and we do get along just fine with our imperfect model of the real world. The purpose of mathematics is to please mathematicians. The purpose of physics is to explain how reality works. Reality suggests avenues of exploration to mathematicians and the results of those studies often solve problems in physics. Reality does not and cannot contradict mathematics. Yes the universe is granular and not continuous. But the granules are so small that most of the time a continuous approximation is accurate enough to get more than reasonable results. When it isn't we use a different mathematical model to get results. Yes quantum physics has contradictory behavior. Our model is good but it still needs work. That has been true for every model that was ever used in physics. Yet they keep on improving it and the model is better now than it was before. It will be even better in the future. I too cannot believe that you are a mathematician.

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I had a problem taking you seriously when you began with "I am a mathmeticians" and you wanted to get serious "respondents" instead of responses. So sorry, no, I do not take you seriously.

Hockey

Unfortunately our education system shields us from the illogical aspects of science. Otherwise relativity and quantum physics might be taught at school rather than university. Having a flat spacetime or a flat Earth would make life simpler to understand but that does not mean a curved geometry has to be illogical. The Flat earth physics might just need to be modified slightly. In most of our everyday situations Flat Earth, Newtonian, physics is a close enough approximation. Rather than Math being worthless, our base assumptions about the world may be wrong. Euclid applied to extra-dimensions produced real world predictions leading to string theory. Playing around with spacetime geometries can produce time warps opening the possibility of paradoxical time travel. It is unclear whether such geometries are inherently self contradictory or whether the outcomes only contradict common sense. Certain configurations of cosmic strings can produce curved spacetime situations with less than 360 degrees in a circle supposedly result in situations similar to travel backwards in time. Such situations defy common sense. This may mean that they are prevented by other aspects of the mathematics or that strange things like time travel are possible. It may turn out that mathematics is self contradictory or that real logical physical worlds are impossible. Deciding if this is the case is a big part of my investigations. It may turn out that we need to work within or around these contradictions, rather than giving up on logic entirely.

Graham P

The term "legitimacy" dose not, in my opinion, apply to any aspect of mathematics (except perhaps some heuristic approaches in applied math). Taking the Peano axioms and the standard deduction rules as given, one can create a rather impressive edifice of theorems, corollaries, and so on that are interesting in their own rite and need not satisfy any criterion of legitimacy. The topology of the plane along with its Euclidean metric has had spectacular success at describing nature but that is not the only reason for its existence. It is intersting in its own right, as are Non-Euclidean metrics on non-Eulcidean spaces, none of which need to satisfy any criterion of legitimacy and many of which are absract structures with no specific representation in "reality". So, I would answer by pointing out to you that whether or not a mathematical structure reflects some reality is simply irrelevant. That the Planck constatnt places limits on what we can measure has nothing to do with pure mathematics at all.

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