Why mathematics is able to predict and describe physical events?
Why mathematics (an abstract thing, according to most) is able
to predict and describe physical events. Why don’t crystal balls, tea leaves, or any number
of other things work? At least they are physical entities!
This problem was famously posed by Eugene Wigner in 1960 in a paper called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” He put the puzzle in strong terms: “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it”.
For example, there are branches of mathematics that allow us to predict the behaviors of planets, comets, missiles, and even certain elements of social systems, such as traffic flow patterns and queuing behavior. Why are the laws of physics so wellcouched in the language of mathematics? Why do mathematical structures find such fruitful application in physics?
There are two kinds of application we can think of here, one more ‘unreasonable’ than the other: (1) a ‘physics-dependent’ kind, and (2) a ‘physics-independent’ kind. For example, there are cases where mathematics has been developed hand-in-hand with some piece of physics: the calculus was constructed with a physical problem in mind, namely how can we solve equations of motion (how do we know how a system will evolve in time). John von Neumann’s creation of Hilbert space (in which quantum states are represented) was also of this kind: a case of finding appropriate mathematical tools for the job. The effectiveness of mathematics is clearly not so unreasonable in such cases: it is a criterion of successful tool-finding that it be an effective one. There were many tools not fit for purpose that were discarded.
The unreasonable kind is that application of mathematics in science (Wigner has in mind physics, primarily, but it generalizes to any mathematically modeled subject matter) that was created independently of physics and yet later found application in physics. Hence, the mathematicians were busying themselves with some purely mathematical problem, not caring a jot about the world of physics, and yet lo and behold this piece of mathematics is found to be a perfect fit for (some aspect of) the physical world. For example, complex numbers find a perfect home in quantum mechanics, as we will see, despite the fact that complex numbers were developed hundreds of years before quantum mechanics was conceived. Non-Euclidean geometries were found to provide the perfect framework for general relativity. The so-called ‘spinors’ of Henri Cartan, from 1913, were found to fit perfectly the intrinsic spin of electrons discovered in 1926 (and were pivotal in the theoretical prediction of anti-matter by Paul Dirac, who combined spinors with the mathematics of quantum mechanics). Wigner’s own example involved a pair of old friends discussing one friend’s job as a statistician. The other friend is incredulous to see that Ď€ (something to do with the ratio of circumference of the circle to its diameter) is appearing in a discussion of the population (humans!) via the Gaussian distribution. How can this be?
There are various responses one could give. One possible response is that even the ‘purest’ (most physics-independent) mathematics comes from worldly investigations at some level, as Stanislaw Ulam points out:
Even the most idealistic point of view of mathematics as a pure creation of the human mind must be reconciled with the fact that the choice of definitions and axioms of geometry – in fact of most mathematical concepts – is the result of impressions obtained through our senses from external stimuli and inherently from observations and experiments in the ‘external world.’
This doesn’t really explain how mathematics can extend physics to go beyond what we can obtain through the senses: our senses are often wrong, and certainly can’t put us directly in touch with atoms and quarks.
Better, one might deflate the puzzle by showing how it is no more miraculous than finding, e.g. a piece of furniture that fits ‘just so’ into some space in a room – and to think, the manufacturer had no idea about my room! I don’t wish to suggest that mathematical physics is on a par with interior design, of course, but in terms of the basic principle behind the deployment of mathematics in physics, I think there are parallels here. But there is a crucial aspect left untreated: the furniture fits because the space has a structure with the right dimensions. It is a matching of (aspects of) their structures that grounds the fit. What about the case of mathematics and physics?
To answer this we might adopt the view that since mathematics is a science of patterns and structure and the world is patterned and structured, there is no mystery about their relationship: they simply have the same structure. Quite naturally, some structures will match up and others won’t: the unreasonable effectiveness then just amounts to finding isomorphic structures, and why should it matter that the mathematical structure was discovered before the physical structure? Further, in physics, one is often dealing with a very limited list of physical features to be represented mathematically, so a matching between them is not so far fetched.
On this view, when one has a match between some piece of mathematics and some aspect of physical reality, then one has made a discovery that the world has this mathematical structure. One can then perhaps explain why the mathematics can lead to surprising physical discoveries – situations where we appear to get more out of the mathematics than we put in, to use Wigner’s expression. A problem with this view is that sometimes mathematics is effective without us wanting to say that there is some structural isomorphism between the mathematics and the world. For example, we might have a model that would be physically inconsistent (such as the early model of the orbit of the electron around the nucleus, which predicted a rapid collapse of the atom) and so couldn’t possibly be matched by reality. There are also all sorts of ‘idealizations’ in physics in which the mathematical model and the world can’t be seen to correspond in terms of actual structure.
Reference: https://philosophy-of-sciences.blogspot.com/2018/12/the-philosophy-of-physics-dean-rickles.html
This problem was famously posed by Eugene Wigner in 1960 in a paper called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” He put the puzzle in strong terms: “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it”.
For example, there are branches of mathematics that allow us to predict the behaviors of planets, comets, missiles, and even certain elements of social systems, such as traffic flow patterns and queuing behavior. Why are the laws of physics so wellcouched in the language of mathematics? Why do mathematical structures find such fruitful application in physics?
There are two kinds of application we can think of here, one more ‘unreasonable’ than the other: (1) a ‘physics-dependent’ kind, and (2) a ‘physics-independent’ kind. For example, there are cases where mathematics has been developed hand-in-hand with some piece of physics: the calculus was constructed with a physical problem in mind, namely how can we solve equations of motion (how do we know how a system will evolve in time). John von Neumann’s creation of Hilbert space (in which quantum states are represented) was also of this kind: a case of finding appropriate mathematical tools for the job. The effectiveness of mathematics is clearly not so unreasonable in such cases: it is a criterion of successful tool-finding that it be an effective one. There were many tools not fit for purpose that were discarded.
The unreasonable kind is that application of mathematics in science (Wigner has in mind physics, primarily, but it generalizes to any mathematically modeled subject matter) that was created independently of physics and yet later found application in physics. Hence, the mathematicians were busying themselves with some purely mathematical problem, not caring a jot about the world of physics, and yet lo and behold this piece of mathematics is found to be a perfect fit for (some aspect of) the physical world. For example, complex numbers find a perfect home in quantum mechanics, as we will see, despite the fact that complex numbers were developed hundreds of years before quantum mechanics was conceived. Non-Euclidean geometries were found to provide the perfect framework for general relativity. The so-called ‘spinors’ of Henri Cartan, from 1913, were found to fit perfectly the intrinsic spin of electrons discovered in 1926 (and were pivotal in the theoretical prediction of anti-matter by Paul Dirac, who combined spinors with the mathematics of quantum mechanics). Wigner’s own example involved a pair of old friends discussing one friend’s job as a statistician. The other friend is incredulous to see that Ď€ (something to do with the ratio of circumference of the circle to its diameter) is appearing in a discussion of the population (humans!) via the Gaussian distribution. How can this be?
There are various responses one could give. One possible response is that even the ‘purest’ (most physics-independent) mathematics comes from worldly investigations at some level, as Stanislaw Ulam points out:
Even the most idealistic point of view of mathematics as a pure creation of the human mind must be reconciled with the fact that the choice of definitions and axioms of geometry – in fact of most mathematical concepts – is the result of impressions obtained through our senses from external stimuli and inherently from observations and experiments in the ‘external world.’
This doesn’t really explain how mathematics can extend physics to go beyond what we can obtain through the senses: our senses are often wrong, and certainly can’t put us directly in touch with atoms and quarks.
Better, one might deflate the puzzle by showing how it is no more miraculous than finding, e.g. a piece of furniture that fits ‘just so’ into some space in a room – and to think, the manufacturer had no idea about my room! I don’t wish to suggest that mathematical physics is on a par with interior design, of course, but in terms of the basic principle behind the deployment of mathematics in physics, I think there are parallels here. But there is a crucial aspect left untreated: the furniture fits because the space has a structure with the right dimensions. It is a matching of (aspects of) their structures that grounds the fit. What about the case of mathematics and physics?
To answer this we might adopt the view that since mathematics is a science of patterns and structure and the world is patterned and structured, there is no mystery about their relationship: they simply have the same structure. Quite naturally, some structures will match up and others won’t: the unreasonable effectiveness then just amounts to finding isomorphic structures, and why should it matter that the mathematical structure was discovered before the physical structure? Further, in physics, one is often dealing with a very limited list of physical features to be represented mathematically, so a matching between them is not so far fetched.
On this view, when one has a match between some piece of mathematics and some aspect of physical reality, then one has made a discovery that the world has this mathematical structure. One can then perhaps explain why the mathematics can lead to surprising physical discoveries – situations where we appear to get more out of the mathematics than we put in, to use Wigner’s expression. A problem with this view is that sometimes mathematics is effective without us wanting to say that there is some structural isomorphism between the mathematics and the world. For example, we might have a model that would be physically inconsistent (such as the early model of the orbit of the electron around the nucleus, which predicted a rapid collapse of the atom) and so couldn’t possibly be matched by reality. There are also all sorts of ‘idealizations’ in physics in which the mathematical model and the world can’t be seen to correspond in terms of actual structure.
Reference: https://philosophy-of-sciences.blogspot.com/2018/12/the-philosophy-of-physics-dean-rickles.html
About Abderrahmane Al jazairi
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