Has Mathematics, as a describer of natural phenomena, become too powerful?
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For example, the Zermelo-Franko axioms of set theory serve as one of the most common foundations of all mathematics - an infinite family of axioms which describe the existence and properties of objects called Sets. (A set is a collection of objects, which are called the elements of the set. Then follows the set operations, the concept of a relation - from this, the concept of a function, then is axiomatized the set of real numbers, then arithmetic, and upon this real analysis is built.)
We see that math is unlike other sciences - anthropology, psychology, geology, biology, chemistry, and specifically physics - these, hereafter referred to as the sciences of natural reality, or the natural sciences. So called because the objects of study considered under each of these sciences are objects which exist in natural reality, in the world and universe around us.
Throughout physics and the other natural sciences, we see that theories often take the form of mathematics; equations asserted as describing the behavior of some space of variables, and their interactions. For example, Newton's Law of Universal Gravitation is an equation which describes how the force of gravity between two bodies is related to the mass of and distance between the two bodies.
Then it is tempting to believe that most, if not all natural sciences can be encapsulated using mathematics - and indeed, the search for a proper 'axiomatization of physics' is one of Hilbert's unsolved questions. However this is an erroneous leap of logic.
In the natural sciences, a better theory is one which more closely models and predicts observable natural phenomena (using the example above, Newton's law of Universal Gravitation is a good model for and predictor of the force of gravity, but it is not perfect. Gravity can be more accurately modelled and predicted by the equations of General Relativity, hence GR is a better theory). In the natural sciences, we progress by 'honing in' on natural reality - getting closer and closer to some objective truth about the behavior of our universe - but not so in math.
As stated above, a mathematical theory is a set of theorems which follow logically from an arbitrary and predetermined list of axioms. If we change our axioms, our theory can change dramatically. An example of this is the five axioms of Euclidean Geometry - drop the fifth axiom, the 'parallel postulate', and we have a new, consistent mathematical theory. These are the axioms of Non-Euclidean Geometry.
And yet historically we have treated mathematics as a natural science. We begin with some collection of assumptions about how mathematical objects 'ought' to behave, discovered that these assumptions then lead to counterintuitive (yet logically sound) theorems, and have attempted to resolve these logical oddities by either changing our assumptions or outright refusing to 'do the thing which causes the problems'. Then we discover that a certain mathematical oddity is perfect for describing a newly discovered natural phenomena, and we accept it as 'truth' (or at least as 'legitimate math').
We have thus fooled ourselves into believing that mathematics is nearing something 'objectively true' - getting closer and closer to the 'reality' of mathematics, as we have seen with the natural sciences. And our current mathematical theory has been remarkably successful across the board of natural sciences; it seems there is nothing mathematics cannot eventually prove or disprove, and has even been touted as the 'language of the universe'.
I would take a more pessimistic position. I would say that in our attempt to axiomatize natural reality as mathematics, we have created an unnaturally accurate beast for which reality is only an approximation.
I point to the Banach-Tarski paradox, a mathematical theorem which proves that any object can be cut up into finitely many pieces, and, simply by rotating and rearranging the pieces, two identical copies of the original object can be produced. The theorem relies on infinity - the infinitude of points in an object idealized as a subset of real-Euclidean space, that infinitely precise cuts can be made, and that infinitely complex and detailed pieces can be produced and manipulated. It is called 'paradoxical' because it seems to run contrary to what we understand about the nature of real, material objects - that they are not infinitely detailed and are, on some microscopic scale, granular and discrete. Yet there are those who believe a banach-tarski construction could potentially be carried out in the material universe, and that the theorem has deep implications about the behavior of subatomic particles.
At what point does the mathematics stop behaving as a predictor of reality, and start becoming an idealization of a world which does not exist in material?
1."Mathematics as a science is the study of abstract constructions of thought using principles of logical reasoning and deduction." - That's a sweeping statement. Not all constructions are abstract. Most of elementary operations are objectively true. there are some abstractions like notation of irrational numbers and some operations in non standard mathematics.
2."a mathematical theory is a set of theorems which follow logically from an arbitrary and predetermined list of axioms." - yes. and we use mathematics because of its rigor. But saying that changes in axioms change everything that follows doesn't affect theories of physics in the sense that you suggest here . Maths is used as a tool in physics. If new axioms are developed in mathematics, it will only help physicists to draw new conclusions, it will not take us away into some other realm of maths per se because there is no such realm. that's just bad use of language.
3." We have thus fooled ourselves into believing that mathematics is nearing something 'objectively true' " - This statement is meaningless because truth is determined by real world observation. Either something is true or false and only that thing is meaningful. Rest are just semantics. And physicists understand this obvious thing. no one's getting fooled.
4."getting closer and closer to the 'reality' of mathematics" - theories take the form of mathematics but interpretation is not maths, it's physics. For example, taking a number from LHS to RHS in a simple equation of the form, say, a+b=c*d => a=c*d-b. but if it is the equation of general relativity where LHS represents curvature and RHS represents energy-momentum. then b will suddenly become an extra amount of energy and we will need new a theory to incorporate it. so what's a small step for mathematicians can be a giant leap for the physicists.
5."we have created an unnaturally accurate beast for which reality is only an approximation." - mathematical operations don't happen in vacuum, they operate on parameters. so what's real or not is understood with the help of theories, not mathematical operations per se. some of the developments in maths surely create an 'awe' among the scientists but there is no reason to infer that we have developed all tools needed to understand the cosmos and godel's incompleteness theorem is surely an 'aw' among the 'awes'. also it is interesting to know that mice can't get out of a prime number based maze because they don't have the biological capacity to do such computations. in case of non standard mathematics , it is reasonable to say that our understandings might also be limited due to our biological limits because here our intuition also plays a role.
6." there are those who believe a Banach-Tarski construction could potentially be carried out in the material universe, and that the theorem has deep implications about the behavior of subatomic particles." - not necessarily. if observations say that some mathematical operations might be unnecessary, then they are avoided even if they give more elegant picture of the world. here it is important to know that the difference between physical quantities and mathematical objects are being misunderstood and problem lies in the traditional definition of physical. it is an old mechanistic interpretation which has been given up centuries ago( as far back as Newton). For example, no one will say that time is a physical thing in the traditional sense. but it is in fact a physical quantity. it does get affected by mass. so common sense is irrelevant here and therefore it doesn't make mathematics any more mystic.
7."At what point does the mathematics stop behaving as a predictor of reality, and start becoming an idealization of a world which does not exist in material?" - I've already talked about the idealization thing in point 5. and on being the predictor of reality , first let's not get any mystic about maths here( point 6). now, interpretation is a matter of theory. lets say if a very simple and elegant theory gives extremely accurate predictions about highly complex 49 things that are verified through observations, it is reasonable to believe in that theory's 50th prediction. ( a potential case for multiverse theory). also if something explains all possibilities in cosmology , it is as good as nothing, though testable in principle ( e.g., string theory(should instead be called a hypothesis): nothing would fall apart if we don't find x or y....string theory makes predictions for 10^500 possibilities to compactify every parameter into 4 dimensions. so it is not the theory of everything. it is a theory of just anything. practically not falsifiable). again, unification of different forces can imply extra dimensions. this is a question of conceptualization and not maths per se. the theory has to be either an excellent predictor in almost all the cases or it has to be testable at least in principle to be acceptable. maybe some philosophers who think 'i' is literally an imaginary number might not understand the implications of an imaginary timeline which is neither imaginary nor an abstract mathematical construct but a prediction of a highly successful theory. it is not only maths (1+2+3...upto infinity = -1/12) but also nature that has shown us that common sense is irrelevant(a universe from nothing - Zero energy flat universe - mathematically beautiful but also most simple explanation of the universe, supported by observation, not just pure metaphysics or some mathematical game).
8. Is maths the only way? perhaps not. mathematical developments can be ad hoc. an alternate which can in fact help in developing maths itself, e.g. , non standard calculus or even fractals is by creating simple abstract programs and creating algorithms that lead to emergent behaviours that aren't obvious in the beginning and can be understood only through running the program can be made a new subject itself like maths and physics. here the basic goal of the field will be to understand and characterize the computational universe using experimental methods ( as proposed by Steven Wolfram) which might eventually help in making understandable theories of the universe ( it is not same as trying to create a picture of the universe. the difference is so small that it is almost trivial to ignore it ). we are not trying to understand the world as a machine but instead trying to create understandable theories because of our biological limit to have only a classical mechanistic picture of the universe. ( by the way it doesn't suggest there is something non physical. one can differentiate between physical and non physical only if one can define physical. but there's no definition of physical. physical is just anything we understand. common sense is irrelevant.)
For instance, we know it is 'objectively' true that 2 * 3 is the same as 3 * 2, in that a grid of objects measuring 2 by 3 has the same number of objects as a grid measuring 3 by 2. Yet mathematically, that multiplication of real numbers is commutative is nontrivial, and is included as an axiom in the field axioms of the real numbers. Perhaps I clarify my original statement as, Mathematics as a _theoretical_ science is the study of abstract constructions of thought.
Applied mathematics, the math of computation and calculation, is then the tool by which physics examines the universe. However it must be the case that nothing in applied math could be called (mathematically) 'true' if it was not based in rigorous theory. Empirical observations and relationships which arise from the study of physics are then not objectively true statements, since there is only at most a finite body of evidence to support any such claim, but rather empirically-derived formulae are artefacts of the physical universe we live in. Certainly these formulae provide insight into the nature and structure of natural reality, which is the goal of physics as a science - but ultimately it cannot be shown as objectively 'true' without a rigorous underlying mathematical theory. It can only at most be 'a relationship which seems to hold in all or most tested cases'
And I would argue again, yes, changing the axioms changes everything. It either limits what a theory can show to be true or false, or it changes that which is considered 'true' under the theory. A theory begins with axioms, definitions and properties of some class of objects - I agree it is true that changing these axioms does not mirror a change in natural reality, but if we discover some phenomena which behaves similarly to our new axioms, then all the theorems of the new theory must apply to the phenomena. Like the example you used, complex numbers and their usefulness in the study of quantum mechanics. Complex numbers were first axiomatized as an artefact of polynomial roots - calculating the root of the general cubic equation necessitated the existence of a quantity with negative square, the lateral unit i - and after this first treatment as a mathematical oddity, their connections to quantum mechanics were noticed.
Perhaps then theoretical mathematics shines in physics when some physical phenomena or structure behaves as if it were an example of some mathematical construction. ie, asserting that our physical universe is an example of a four-dimensional Euclidean space, or asserting that a certain dynamic system evolves according to this system of partial differential equations. Once such a connection is established, all the theorems underlying the theory of such a structure immediately apply to it, and can make predictions then observable and testable. If the theory predicts something contrary to what we observe, then our initial assumption of what structure our physical phenomena is (or represents) was incorrect - ie, the math is not wrong, our initial interpretations are just off.
I agree interpretation of math is physics, since physics aims to ultimately describe natural reality. However, understanding the subtleties of an equation and the notion of quantities its variables represents does not affect the truth of the mathematical statement. Either this set of variables is related to this other set of variables in this way, or it is not.
The awe that a physicist or mathematician feels when reflecting on the implications of a mathematical theorem is exactly the awe I would warn against - either the theorem holds for observable phenomena or it does not, and if it does then it should not be surprising that the math predicts it so beautifully, when it can be shown to follow logically from the axioms. It is when we believe that the math outside our universe can influence and change the nature of our reality that we lose sight of the role of math as a science.
The role of math in physics, you say, is a tool to model and predict phenomena, as I have said - but specifically, to help us as humans understand the theories better. This is an intriguing proposition, and I would lean towards agreeing with you. Even in math, our abstractions have been constructed based on our intuitions of how some class of mathematical objects 'ought' to behave - and then we go backwards and show that our theory indeed predicts everything we want it to predict. Be that as it may, as an arbitrary artefact of logic and reasoning, then surely at any given point there must exist a set of equations which more closely predicts and models natural reality than those which we currently call 'the Laws of physics' - simply because there are infinitely many such equations which could conceivably be written down. Hence for any system which predicts and models natural reality, there must exist a mathematical system which is a 'better' theory than it, and in fact, infinitely many such systems. The physical and metaphysical interpretation of such a theory must be inexplicably connected, else we separate the mathematics from its meaning.
1,2. I agree i shouldn't have used the words 'most operations' but again talking about what's true or false is meaningless in the context of maths ( Point 3 of my previous round arguments ). if you want to call it objectively true , then your definition of true false is not scientific but rather it means what people mean in daily lives, a non technical and if you look carefully, just a random definition.
3. When we say something is true in science ( say in physics ) we mean our theory is closer to truth. that's the scientific use of true. in science we don't prove things, instead we falsify theories. again when if i say it is true that there is a moon nearby earth, that statement is either true or false and it can be observed. but tautology and contradictions have nothing to do with true or false because it is not giving any information to observe and falsify.
4. your paragraph 4 agrees with what i argued . I see no point of contention. Yes maths gives us new avenues but as i said in the later parts of point no. 8 in my previous round arguments, we make understandable theories of the world in science. and there is no so called realm of maths in its own . that's mere semantics. we see things through the lens of maths and there is no reason to believe that we'll end up with the correct theory but that has also to do with many other things such as loss of important data after the big bang, our mechanistic view of everything and so on. but it nowhere implies maths is taking us to the wrong direction because the onus of interpretation is not on maths.
5. yes it means our interpretations are off but maths doesn't force us to keep that view. we continue to look for observations. as i said in previous round, there can by many theories about the universe having internal logical consistences with beautiful maths. but that doesn't make maths a brute. we don't know if we have all the mathematical tools yet, and since we are doing empirical study, we'll never know that for sure.
6. your paragraph 6 is in agreement with me. plus not affecting 'truth' of mathematical statement doesn't mean we are forced to use certain mathematical form. if it is highly likely we look for evidences. and at the same time we also make alternate theories with different maths. we don't change the tool, we switch tools.
7. "It is when we believe that the math outside our universe can influence and change the nature of our reality that we lose sight of the role of math as a science." - if there are physicists who believe what you say here, then they are simply idiots who are more about since fiction(more like Michio Kaku). then i'm in agreement with you.
8. " there must exist a set of equations which more closely predicts and models natural reality than those which we currently call 'the Laws of physics'"....."or any system which predicts and models natural reality, there must exist a mathematical system which is a 'better' theory than it" . prediction closer than reality? that's semantic hocus pocus. you can say some models are more likely than what observations show, it is for this reason we look for evidences. for example our initial observations showed that ours is an open universe but mathematically it should be a flat universe. so scientists came up the idea of dark energy that also explains why the expansion of our universe is accelerating. but the point here is physicists chose flat universe not only because that's a theoretically more simple explanation(there is no reason to believe that theories of the universe should be ultimately very elegant ) but more importantly because only in a flat universe the total sum of energy can be zero and only such a universe can come out of nothing without any divine intervention. again. mathematical system better than theory? of course there can be. but giving human value by calling them better is a layman thing. for example it would be better if the empty space didn't weigh anything. it would be mathematically more ideal for complex systems to evolve. but observations show that you take out all the energy and all the matter our of empty space and it still weighs something, an observation that took everyone by surprise. also if you go into works on multiverse( no theory yet) , there can be infinite universes with infinite different set of laws. so what you call possibilities better than our world's can be seen in such a theory. maths doesn't operate in vacuum but on certain parameters, if those parameters are reduced to absurdity it is reasonable to restrict mathematical operations.
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