That is an important question. Is mathematics a convenient construct — a human invention — that we humans use to describe nature? Or is mathematics more fundamental — intrinsic to the Universe — mirroring the divine ordered creation of a reasonable logical Creator?
Theoretical physicist Lee Smolin in his book1 “Time Reborn” argues that what he calls the Newtonian paradigm is a myth. What he labels the Newtonian paradigm is the attempt by theoretical physicists, beginning with Sir Isaac Newton, to describe the Universe with immutable laws, using a mathematical description. Those laws are unchanging in time, as reflected in the idea that Newton understood the laws of nature as the result of Divine creation and hence that they are unchanging in time. Newton wrote:2
And from true lordship it follows that the true God is living, intelligent, and powerful; from the other perfections, that he is supreme, or supremely perfect. He is eternal and infinite, omnipotent and omniscient; that is, he endures from eternity to eternity; and he is present from infinity to infinity; he rules all things, and he knows all things that happen or can happen.
From this standpoint he understood the laws of nature as the special creation of God. These resulted concomitant with the special creation of the Universe itself. The Universe is not the result of blind chance acting on some initial conditions and evolving accordance with those laws.
Some have claimed that Johannes Kepler said, “I am merely thinking God’s thoughts after him,” though there is no primary source for the quotation. However Johannes Kepler did write:3
Geometry is one and eternal shining in the mind of God. That share in it accorded to men is one of the reasons that Man is the image of God.
And also Kepler:4
Geometry, which before the origin of things was coeternal with the divine mind and is God himself (for what could there be in God which would not be God himself?), supplied God with patterns for the creation of the world, and passed over to Man along with the image of God; and was not in fact taken in through the eyes.
Both Newton and Kepler held to the notion that the Universe and its laws, which can be described by mathematics, are the product of an Intelligent Mind, a Creator God, existing before the creation of the Universe. And even if the mathematical description a scientists discover is only an approximation to some far more complex mathematical and more complete description it does not undermine the validity of using mathematics to describe the laws of nature. This is noted by the improvement that Albert Einstein made to Newton’s theory of gravitation. Einstein’s General Theory was an extension on Newton’s and included it over the range of operation of its applicability, i.e. when the gravitational forces are weak, and limited in range. That has been experimentally verified in the laboratory.
Lee Smolin, however, sees the Newtonian paradigm as a timeless description of a block universe. That is, the laws are perfect and immutable in time. They do not change over time. The Newtonian paradigm then, he says should apply to the whole universe, but it fails. Hence Smolin proposes small random changes in the laws of physics over time, particularly as his imagined universes evolve from the black holes that spawned them in their parent universes. His is a fanciful scheme of cosmological natural selection generating new universes. His belief then is there is no intrinsic law, no perfect mathematical description of the Universe, because there is no Creator. He writes:5
Because the Newtonian paradigm cannot be scaled up to include the universe as a whole, there need be no mathematical object corresponding to the exact history of the entire universe.
Stephen Hawking asked in his book “A Brief History of Time,”6
“What is it that breathes fire into equations and makes a universe for them to describe?”
To which Smolin responds:7
Such utterances reveal the absurdity of the view that mathematics is prior to nature. Math in reality comes after nature.
This means of course that Smolin does not believe in any intrinsic fundamental immutable law, which the mathematics describes. He goes onto to write:7
There are aspects of the real universe that will never be representable in mathematics. One of them is that the real world is always at some particular moment.
He believes time is real in the sense that the universe evolves, hence the laws of physics evolve. Because we always exist in the present moment he reasons that this is not describable by a timeless law. He writes:7
The universe simply is — or better yet, happens. It is unique. It happens once, …
But the “question of why the universe exists rather than not” he cannot answer. He falsely states that its origin from nothing cannot be answered by the existence of a Creator God, because that would mean God existed prior, i.e. God is something which preceded the Universe, and hence the Universe did not come from nothing. But God is the great I AM who always existed, the uncaused Cause. He had no material causation.
In regards to idea that mathematics is needed or not to describe the Universe, Smolin claims that in some cases mathematics is not needed. He gives as his example, natural selection.8
…aspects of it have been captured in simple mathematical models, but no single model captures the whole variety of mechanisms by which natural selection acts in nature. Indeed, new mechanisms of evolution may emerge at any time, as new species are born.
Of course his statement is a faith assertion, nothing more. He does not back it up with any scientific rigor, certainly not in regards to natural selection as it really operates in nature but with an appeal to Darwinian goo-to-you evolution.
In the following we shall take a brief look at the nature of mathematics as applied to what is now called Geometric Algebra.
You may recall being introduced to algebra in high school. That algebra involved only scalars, quantities with no direction associated with them. We call them dimensionless, like a point, a dot on the page. Then later you may have studied vectors where a direction was added. The resulting object is a line, we call 1-dimensional. Now mathematical developments have taken those ideas much further, developing the bivector, a plane, a 2D object and the trivector, a volume, a 3D object. These objects together form the basis of the field of mathematics, formalised by William K. Clifford, but more generally now known as Geometric Algebra.
That algebra is a closed finite universe that is able to generate time from the properties of space itself.9
One blogger writes:10
Mathematics is not a human invention but more of a discovery…. The properties of math are the direct evidence for the principles of operation of the mind and brain.
He believes this because he sees in Geometric Algebra (also called Clifford Algebra) a description that implies mathematics is an artifact of the human mind and how it represents spatial reality. He says:
It is a unifying language for mathematics, and a revealing language for physics.
Below I quote (in blue text) from his writings with my emphases added:10
The history of algebra has been the history of an incremental discovery of an evidently pre-existing structure. Clifford Algebra has turned out to be the lynchpin that shows how all the previous discoveries are interrelated within a single self-consistent scheme whose laws operate the same in all directions. It is the very simplicity and generality of Clifford Algebra that confirms its “truth”. But truth to what? What is it that mathematics represents? What must it remain true to? This gets to the question of, as Lakoff and Núñez (2000) called it in the title of their book, “Where Mathematics Comes From“. If algebra is the discovery of a pre-existing structure, then what is that structure and where is it located? Plato and his many modern adherents believe mathematics to be an objective eternal Truth that exists independently of human minds, although it is only accessible through the human mind.
It is true that mathematics is not a human invention, but more of a discovery, and thus has objective existence in that sense. But to declare the magnificent edifice of Mathematics to inhabit an orthogonal dimension that is inaccessible to scientific scrutiny, is a hypothesis that cannot in principle be falsified, and thus, it is not a scientific hypothesis but more of a belief, for those who are inclined to believe it. I demand a more scientifically sound falsifiable hypothesis for the ultimate nature of mathematics that gives it objective existence in the universe known to science. I agree with Lakoff and Núñez that mathematics is an artifact of how our mind makes sense of reality, and that therefore mathematics has physical existence or instantiation within the human brain. Our view of the world is not direct, we see the world indirectly, through a representation, or model of the world constructed in our brain, and that model is painted out in the geometric primitives of points and lines and planes, which are the elemental features of perception, and those geometric primitives correspond closely to the algebra which was ultimately derived from them.
Since Clifford Algebra offers such a clear view of the essential principles of mathematics, it thereby also offers a clear view of the principles of visual perception and the way we conceptualize shapes in space, revealing algebra to be at the root of it a spatial process of spatial operations that operate on spatial structures. The scalar algebra that we learn first in school is an abstraction of the vector algebra that underlies it.
The periodicity of the number line, the alternation between even and odd, the concepts of negation and inversion, are all strongly suggestive of a phasic / cyclic oscillatory phenomenon which must ultimately reside in the brain. The fact that Clifford Algebra reveals all of algebra to be a spatial phenomenon at the root of it, suggests that the computational processes of the human brain are themselves spatial in nature, like the computational operations of nonlinear optics and phase conjugation which they resemble, where spatial waves interact with other waves to produce modulated waves. The fact that Clifford algebra is particularly good at modeling phasic / cyclic phenomena such as the “spin” of elementary particles, all suggests that the human brain operates with a spatial representation based on cyclic oscillations and standing waves in the brain, which just happens to be similar to the cyclic resonances of subatomic particles. The equivalence found in Clifford Algebra between the linear and rotational domains, how negation maps also to rotational reversal, and translation along a linear axis maps to angular rotation about the origin, is surely further evidence for a wave based principle of computation in the brain.
The many manifestations of closure seen in different forms throughout algebra suggests a computational principle that makes optimal use of a finite representational resource, like the odometer whose individual dials each form closed loops, as does the odometer as a whole, and thus the mechanism never “runs off the end”, it just wraps around again. This is not a strategy for an accurate representation of an effectively infinite universe, as we discover whenever our odometer “rolls over” to zero, but rather it is a strategy for a wise design of a limited representational resource. It is an artifact of the computational mechanism of our numerical mind with which we represent the world, not a property of the world itself. But there is also a certain invariance to a rotational representation that resolves the problem of the “infinities” found at both ends of every number line. The equivalence between linear and rotational dimensions suggests that our mind never actually uses linear dimensions like the X axis, but approximates them in a circular dimension to avoid the problem of “infinities”.
This entire analysis of Clifford Algebra was based on my own foundational assumption that mathematics is not a human invention, but more of a discovery of the essential principles of computation in the brain. If this hypothesis is correct, then it should be possible to reverse engineer the principles of operation of the brain with guidance from the properties of mathematics as revealed through Clifford Algebra. … When we discover what kinds of things can be computed using a cyclic / phasic analog system built to emulate Clifford Algebra, only then will we be ready to hypothesize the computational function of the oscillations found in the brain.
This is his hypothesis. It suggests that mathematics is not only intrinsic to the Universe but fundamental to the human brain. Add to that the idea that we humans were created in the image of God, our Creator. This tells us the following. The mathematical description of the Universe, though immensely complex and still largely unknown (Ecclesiastes 3:11), is not an invention of man, but a creation of God. And when we try to discover the mathematical description of the Universe, which our mind was created to be able to understand, we are “thinking God’s [mathematical] thoughts after Him.”
- Lee Smolin, “Time Reborn: From the Crisis in Physics to the Future of the Universe,” Mariner Books, New York, 2014
- Sir Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy (1687), 3rd edition (1726), trans. I. Bernard Cohen and Anne Whitman (1999), General Scholium, 941.
- Johannes Kepler, Conversation with the Sidereal Messenger [an open letter to Galileo Galilei], Dissertatio cum Nuncio Sidereo (1610), in Johannes Kepler Gesammelte Werke (1937- ), Vol. 4, 308, ll. 9-10.
- Johannes Kepler, Harmonice Mundi, The Harmony of the World (1619), book IV, ch. 1. Trans. E. J. Aiton, A. M. Duncan and J. V. Field (1997), 304.
- Smolin, op. cit., p. 245.
- Ibid., p. 246, as quoted by Smolin.
- Ibid., p. 246.
- Ibid., p. 247.
- J.M. Chappell, J.G. Hartnett, N. Iannella, D. Abbott, Deriving time from the geometry of space, preprint available here http://arxiv.org/abs/1501.04857
- slehar, Clifford Algebra: A visual introduction