The Nature of Mathematics
By Rich Van Winkle
Mathematics is the scariest of subjects – not because of its complexity, but because it forces us to examine our own simplicity.
What follows is a non-technical analysis of the nature of mathematics. Yes, there is some simple math and I have provided examples which are not necessary for grasping this content. If you don’t excel at math, just read the text and ignore the examples.
A few thousand years ago, it started out simply enough: counting 1,2,3,4,5,6,7,8,9,10… Then addition 1 + 1 = 2; 1 + 2 = 3… Then multiplication 1+1+1+1 = 4*1… 2+2+2 = 2*3… Then, oh yes, you can do these things forwards and backwards. It didn’t take long however before someone started asking deeper questions and in doing so they “discovered” that the numbers and processes (addition, subtraction, etc.) of mathematics (just “math” hereafter) could describe or define seemingly real things – circles, squares, triangles. And so, math soon included geometry (math’s relationship to shapes). It didn’t take long for some pretty smart people to see that there was something weird and special going on with math.
"Geometry existed before creation." – Plato
"The book of nature is written in the language of mathematics." – Galileo
This short description of the nature of mathematics is my attempt to explain why such great thoughts reveal a greater truth and why our approach to teaching math has hidden this greater truth. My approach is one of re-definition and re-application. Consider this…
"We know that God exists because mathematics is consistent…”- Andre Weil
What the heck is mathematics? In school, they taught us that math is a way to solve problems… problem after problem after problem. Somehow our math teachers missed its essence and meaning and, by making it a boring methodology, hid its importance (unless we had interest in some field that needed its problem-solving potential). I mean, did any math teacher ever start a class by telling us:
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty." - Bertrand Russell
I don’t know about you, but I never saw the “truth” or the beauty in long division, equations, set theory, finding the area of a circle, trigonometric functions, or derivatives. It was nice to find “short-cuts” in solving certain engineering problems, but that fell far short of inspiring me. In fact, it never even occurred to me that it SHOULD inspire me. Then, along came Einstein. Well, actually, he had already been around and was no longer with us, but his work on relativity entered my mind like a beautiful sunset. The math was largely beyond my skills, but the logic of it made sense – wonderful sense.
In for a dime… I moved into quantum physics and my views of reality and divinity haven’t been the same since. The real purpose of this work is to explain why.
We should, of course, start at the beginning. Numbers are not really mathematics, but are essential to its functions. Numbers began as just a symbolic representation for quantities (“natural numbers”). Because of the biological imperative of having ten fingers, we adopted a “base” of ten for our number system (and somebody figured out we needed a zero to make the system work). Have you ever considered it odd that we need two digits (a one and a zero) to represent the final finger when we count them?
It’s pretty intuitive to see that we needed a way to represent large numbers with fewer symbols and the decimal system works well for this. But we tend to think of decimals as partial numbers - like .25 meaning 25/100 or one fourth of something. Fractions are probably even more intuitive than decimals – something we figured out the first time we needed to divide a pie. These numbers (and negatives) are known as “rational” numbers and they are human representations of real things. We developed a “shorthand” for representing multiple multiplications of the same number (raising to a power or exponentiation such as 24 = 2*2*2*2 = 16) and that led to an inverse operation known as “rooting” (as in square root, cube root, etc.). Then something rather weird happened – we had to wonder what the result of or might be.
So what number times itself yields the value of 2? Try 1.4142135623730950488016887242096980… Try as you might – even with 102000 digits – you won’t get exactly 2. This is still a “real” number, but it seems to not make sense. Perhaps that explains why it’s called an “irrational number” (even if that makes sense). The square root of two requires “infinite” digits – meaning it is open-ended or unlimited. The whole concept of infinity is rather counter-intuitive and is well explained by one of its earliest graspers - Zeno of Elea. In one of Zeno’s paradoxes, he poses this problem…
Suppose that you want to walk to the store. To get there, you must first get halfway there. Before you can get halfway there, you must get a quarter of the way there. Before traveling a quarter, you must travel one-eighth; before an eighth, one-sixteenth; and so on. Because this division may occur to infinity, it is impossible to reach the store in a finite amount of time.
The paradox demonstrates a problem of “truth” – the premise is true but the reality seems to disprove it – we can reach the store in a finite amount of time (usually). It would seem that mathematics demonstrates that all motion must be an illusion or that motion (involving both time and distance) is not actually divisible. The very existence of both real and mathematical infinities is one of the most revealing parts of reality.
I do not wish to get distracted by such deep issues; we have deeper ones to deal with. Mathematics has proven uncanny at explaining relationships within physical reality. Or, as Einstein put it,
"How can it be that mathematics, a product of human thought independent of experience, is so admirably adapted to the objects of reality?”
Nowhere is this odd relationship between math and objective reality more evident than in the ratio of a circle's circumference to its diameter – π (“pi”). While mathematically complex, this simple ratio expresses a fundamental constant built into the very fabric of the universe. As it turns out (pun intended – or “PI”) this ratio emerges often in surprising relationships throughout science (e.g. number theory, statistics, thermodynamics, mechanics, and electromagnetism). It is also viewed as a mystical number as should any transcendental number (consider that each of the transcendental constants contain more information than the universe itself).
Before we move on, we should briefly note two other mathematical constants - the base of the natural logarithm (“e”) and the “imaginary” number (“i”). These constants are a bit more obscure than π, but exemplify something equally important. The natural logarithm (represented by the symbol “e”) of a number is the power to which the natural logarithm would have to be raised to equal that number: The natural log expresses the relationship between addition and multiplication (and served as a basis for slide rules)… Natural logs were first related to exponential growth but have important application in probability theory (“normal distributions”) where it arises in a subtle way not related to the sum of an infinite series (as in compound interest). The constant “e” (Euler’s constant) represents the limit of (1 + 1/n)n as n approaches infinity and is another transcendental number.
Finally (bear with me), we should quickly note the problem of . Since there is no real number which may be multiplied times itself to produce -1 (i2 = −1), the answer is deemed an imaginary number. But when the expression is raised to the power, the result is an infinity of values that are surprisingly all real. What?
OK, if you’re not on overload, you’re probably not getting all this. And you’re probably thinking – what the heck does that have to do with anything REAL? The answer is simple – trust me. But first, answer this question: If you start from a particular point and walk forward 5 steps, turn left 90˚and repeat three time, where are you? (Lost?). You’re right back where you started! If you were to diagram this, it might look like this…
We do this without thinking of it as a two-dimensional action (up-down/right-left). We also tend not to view this as rotation around an axis…
And, we don’t normally think of our turns as positive or negative: a right turn being positive and a left turn being negative, but if we use the axial representation where the upper side of Dimension One is positive and the right side of Dimension Two is positive (the opposites being negative or minus according to our axis), then we can represent our rotation as follows:
The first 90˚ rotation is positive (“turn 1”)
The second 90˚ rotation is negative (“turn 2”)
The third 90˚ rotation is negative (“turn 3”), and
The fourth 90˚ rotation is positive (“turn 4”).
The two positives and the two negatives cancel each other out and we end up back where we started. But wait, the positive rotations can be represented by real numbers and the negative rotations by “imaginary numbers” and the mix of the two gives us “complex numbers”. Complex numbers somehow represent rotation in two dimensions - Multiplying by a complex number equates to rotating its angle. Weird!
And now we should return to the actual topic at hand – the “nature of mathematics”. So far we’ve only looked at numbers, but we should already have an inkling of one part of the picture: numbers represent real things as a convenient way to figure out quantities. But there are certain numbers (“constants”) which seem to arise from the very nature of reality – the relationship between physical structures (π) and the relationship between numbers (e). These “transcendental” numbers are really unexpected and we simply cannot say why pi π isn’t simply equal to 3 instead of 3.141592653589793238462643383 2795028841971693993751058209749445923078164062862089986280348253421170679… And, we know that there are different kinds of numbers which are needed to properly describe reality. For now, I will merely say that numbers themselves tell us something important about the nature of reality.
The next aspect of math is operational or functional. A math operation is an action or procedure which produces a new value from one or more input values, called "operands" (i.e. and addition, subtraction, negation, multiplication, division, and exponentiation). Operations can involve mathematical objects (such as logical operands, rotations, vectors, and sets) as well as numbers. Numerical operations form the basic realm of math known as arithmetic.
A math function is a relation between a set of inputs and a set of permissible outputs where the property that each input is specifically related to one output. We generally give functions names in a manner akin to the way we name objects that have a specific relationship. For example, if a tree grows 2 feet per year, we might create a function: h(age) = tree’s age in years × 2. Functions are a good indication that math has language-like aspects and they might be viewed as being similar to sentences.
Algebra is the application of and rules for manipulating symbols using operations and functions. Meaning "reunion of broken parts", Algebra provides uniform methods for solving equations (an expression of a formula). Because algebra uses variables instead of numbers, its computations allow for proofs of properties that are true regardless of quantities. In other words, algebra provides a means for proving “truth” independent of the specific reality. But more importantly, it tells us something profound about the Universe.
The critical part of any equation is equality (Duh!). When we say that 1+1=2, we are symbolically representing the quantitative sameness of both sides of an equation. That should make perfect sense, but then we look at something like this…
Hopefully, this makes no sense to you (just kidding). But, if this makes sense to you, then you probably know more about math than I do. But you don’t need to be an astrophysicist to see something critical here – the equal signs are telling us that these things are equivalent and that they solve to a value of ZERO. Hmm. This leads to several questions:
1. Why are so many critical aspects of physical reality mathematically equivalent (sameness)?
2. How can so many of these aspects ultimately equate to nothingness?
3. Why do many others equate to unity (one)?
We have entered the era of cosmology and the quantum. Is it not fascinating how much the field of the largest (the Universe) and the field of the smallest (sub-atomic) have become linked? Cosmology reveals the patterns, structures, and interactions behind physical reality. Quantum mechanics reveals that those patterns, structures, and interactions are indeterministic and just simply weird. Both fields have largely been derived or discovered from mathematics.
Quantum indeterminism is derived primarily from the work of Werner Heisenberg and his heuristic argument that the more precisely the position of any quantum particle is determined, the less precisely its momentum can be known (and vice versa) is a fundamental property of quantum systems. The explanation for this property is derived from the following formula:
If you must know, this formula says that the position of a quantum particle is described by a wave function which is interpreted as a probability density function. The position of a particle is uncertain because it could be anywhere along the wave packet. As the wave function is an integral over all possible modes, a Fourier transformation reveals that the momentum must become less precise (as the mixture of waves has many different momenta). But wait!
We may quantify the precision of the position and momentum using standard deviation methods to show that there is a probability density function for position. So, it’s not bad enough to say that we simply cannot know the EXACT position and momentum of any quantum particle at any moment, we have to add that these basic properties are controlled by probability. Thus, we cannot know both the exact position and exact movement of any quantum particle, but we can determine the probability of these properties. What this math tells us is that the dynamic Universe unfolds based upon a combination of chance and controlling laws.
Let us look quickly at a few of those controlling laws.
First, we might want to understand the relationship between two of the most basic aspects of the Universe: space and time. Once thought to be distinct and invariant, space and time were unified by Einstein under “special relativity”. This wide-ranging theory (now accepted as among the most proven theories in science) yields many unexpected (and experimentally verified) consequences, including length contraction, time dilation, relativistic mass, a universal speed limit, and relativity of simultaneity. But the best known and most revealing prediction of special relativity is a simple formula: E = mc2. The equivalence of energy and matter (under the factor of the speed of light squared) is a compelling revelation. All those objects that seem so “solid” and permanent are just packets or collections of energy acting in accordance with probability. Einstein’s math reveals that reality is not what it seems.
And, special relativity leads us to general relativity, Einstein’s theory of gravitation…
Like special relativity, the theory of general relativity yields unexpected consequences like gravitational time dilation, gravitational lensing, the gravitational redshift of light, and the gravitational time delay (all of which have been repeatedly confirmed by experiments). General relativity also suggests the existence of “black holes” and gravitational waves which have both become observed and studied. What general relativity tells us about the Universe is equally weird – space and time are not separable.
Under general relativity, gravity is shown to be a geometric property (represented fully by mathematics) of spacetime. Spacetime is shown to be a Universal substrate which may be curved or distorted in a manner which directly affects the energy and momentum of any matter or radiation which is present. Since we have already shown that the properties of energy and momentum are probabilistic (mathematical), we may say that spacetime is a probabilistic geometrical structure which underlies the Universe. That structure is modelled from a system of partial differential equations. (“God Integrates Empirically.” AE).
Along with relativity, we know much about the background structure of the Universe in the form of physical laws and constants – each of which is fully represented by mathematics.
The physical laws of the Universe are readily divided into categories and these categories are best identified by mathematical similarities. Thus, the laws of conservation can be expressed using a general continuity equation:
(ρ = some quantity per unit volume, J is the change in quantity per unit time per unit area).
Since the rate of change of density in a region of space must equal the amount of flux leaving or gathering in any isolated region, there is a constant or conservation of mass/energy.
The laws of mechanics, including Newton's laws, Lagrange's equations, and Hamilton's equations, may be derived from a single mathematical representation:
Motion and mechanical interactions are thus readily predicted using math (from quantum to interstellar – some using slightly more involved representations)). Incredibly, the entire dynamics of the Universe for all times is reducible to mechanical laws – coupled with the probabilities inherent in physical interactions.
Next we should consider the laws of thermodynamics – laws which reflect the energy interchanges which occur routinely within the Universe. “All things in the observable universe are affected by and obey the Laws of Thermodynamics. In the “First law of thermodynamics” we find that any change in internal energy in a closed system is accounted for entirely by the heat absorbed by the system and the work done by the system. This is represented by this equation:
This tells us that matter/energy cannot be created nor can it be destroyed. The total quantity of matter/energy within the Universe must remain constant. The “state of the matter may change (.e.g from solid to liquid to gas and back again), but the total amount of matter/energy remains the same.
In the “Second law of thermodynamics” we find that the entropy of isolated systems never decreases OR that any reversible change has zero entropy change (as irreversible processes are positive). This formula is written as:
This simple formula represents a rather complex result: while the quantity of matter/energy must remain constant (First Law – as above), the quality of matter/energy will deteriorate gradually over time (within a “closed system”). Good usable energy is inevitably used in the production of more complex systems, for growth, and during “repairs”, but such gain always results in the conversion of more usable energy into unusable energy – “good” energy irretrievably becomes less usable “bad” energy. This is commonly known as entropic conversion since "Entropy" is defined as a measure of unusable energy within a closed or isolated system (such as the universe). Any conversion of usable energy into unusable energy increases "entropy". This law tells us something essential about the universe and our odd circumstance within it (as below).
Entropy may also serve as a gauge of “chaos” (disorganization or randomness) within a system. If entropy will always increase in the Universe, then any exception requires that good energy must be lost somewhere else and that over time, it will become balanced at the lower level throughout the Universe.
The implications of these laws transcend the mathematics behind them and lead to fuller discussion elsewhere. For now, we will move on to other physical laws…
Our next category of physical laws deals with electro-magnetism and are generally grouped into “Maxwell's equations”. Yes, the dynamic distribution and evolution of the electric and magnetic fields in relation to electrical charge and current throughout the Universe is represented by mathematics. Grouped in “Maxwell’s equations are:
Gauss's law for electricity:
Gauss's law for magnetism:
Faraday's law for induction:
Ampère's electrical circuit law:
(with Maxwell's correction)
Lorentz”s law (for point charges due to electromagnetic fields):
Schrödinger’s equation describing the time dependence of a quantum mechanical system:
Planck–Einstein law (holding that the energy of photons is proportional to the frequency of the light):
(the “h” here is Planck's constant).
This leads us (naturally) to discussion of the physical constants…
The very idea of a natural “constant” seems odd – the Universe is clearly a dynamic and evolutionary system where change is the norm. Where does something that never changes enter the system – and WHY? I love the way that Richard Feynman expressed it when considering the physical constant “e” – the amplitude for a real electron to emit or absorb a real photon …
“There is a most profound and beautiful question associated with the observed coupling “e”... It is a simple number that has been experimentally determined to be close to 0.08542455… It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!”
We might also wonder what a “constant” has to do with mathematics. Obviously, the physical constants are quantities and we represent quantities using numbers. Less obvious is the fact that many (most) of the constants represent some relationship between physical features of the Universe and several seem to fill a mathematical NEED. Huh? We’ll get to that after a short overview…
There are plenty of physical constants and we divide them into two groups: dimensional physical constants that are dependent on the units used to express them and dimensionless physical constants (aka “fundamental physical constants”) where their numerical value is the same under all possible systems of units. In other words, dimensionless physical constants cannot be derived from any other source and can only be measured from nature. Currently there are 26 known dimensionless physical constants, including:
· the fine structure constant (α = 7.297351(11) x 10-3)
· the coupling constant for the strong force (αs ≈ 1)
· the gravitational coupling constant (αG ≈ 10−38 in Planck units)
· the masses of the fundamental particles
· the cosmological constant (λ ~ 10−122)
· the ratio of the fine structure constant (the dimensionless coupling constant for electromagnetism) to the gravitational coupling constant
· the number of spatial dimensions (~3-13)
A partial list of the dimensional physical constants includes…
Symbol Meaning Value
c Speed of light in vacuum 2.997924562(11) x 1010 cm sec-1
h Planck's constant 6.626196(50) x 10-27 erg s
k Boltzmann's constant 1.380622(59) x 10-16 erg K-1
e Elementary charge of an electron 4.803250(21) x 10-10 ESU
me Rest mass of the electron 9.109558(54) x 10-28 g
G Gravitational constant 6.6732(31) x 10-8 dyne cm2 g-2
NA Avogadro's number 6.022169(40) x 1023 mole-1
e / m Electron charge-to-mass ratio 5.272759(16) x 1017 esu g-1
R Rydberg constant 1.09737312(11) x 105 cm-1
a0 Bohr radius 5.2917715(81) x 10-9 cm
R Gas constant 8.31434(35) x 107 erg K-1 mole-1
Stefan-Boltzmann constant 5.66961(96) x 10-5 erg cm-2 s-1 K-4
au Astronomical unit 1.49597892(1) x 1013 cm
eV Electron volt 1.23963 x 10-4 cm
Since we already noted one important physical constant – π (pi doubles as a mathematical constant), we may as well start with it.
A constant may be viewed in different ways; either as something which is simply inherent and emerges as a result or something which is essential in the design or operation of a system. While either is possible within the Universe, there are compelling reasons to see the physical constants of the Universe as being elements of creative design. This is most evident when we see how the constants fit within the framework of mathematics.
Pi, for example, transcends its pure mathematical role as an essential value in either defining or applying other physical constants. The dimensionless fundamental constant known as the fine structure constant is thus represented by this formula…
(Where e is the elementary charge, ħ is the reduced Planck's constant, c is the speed of light in a vacuum, and ε0 is the permittivity of free space. The fine structure constant is fixed to the strength of the electromagnetic force).
The ratio of a circle’s circumference to its diameter is specific to the dimensional nature of the Universe and cannot be defined within the Universe (as it is infinite and the Universe is finite). Thus, we should view pi as representing something “behind” the Universe (as arising from the substrate upon which the Universe is unfolding).
This leads us to consider the most basic and often overlooked fundamental constant – the number of dimensions in the Universe. In everyday life, we experience four dimensions: the three “spatial” dimensions (up-down, right-left, front-back) and one unidirectional temporal dimension (“time”). While handy in description, we now understand that spatial and temporal dimensionality within the Universe is really just one thing (a true “singularity”). Our perspective on and experience of these dimensions is largely an illusion. The Universe has and supports dimensions which we do not routinely experience. Whoa!
In physics, a "dimension" refers to the mathematical description of the structure behind the Universe. I like to think of dimensions as the way the Universe keeps things separate. We accept that no two objects may occupy the same space at the same time (how confusing would things be otherwise). We believe that events occur in sequence (time moves continuously in the same direction). But both of these assumptions are only true within the space-time dimensionality of our sensory awareness. We never experience time stopping, slowing, or moving backwards. We never encounter a space with two objects overlapping. Thus, we have assumed that our dimensions are also those of the Universe. Quantum physics proves (or very strongly indicates) otherwise.
A full discussion of dimensionality is beyond our scope here, but the mathematical nature of dimensions and the ability of mathematics to work beyond the “basic dimensions” of space-time is one of the key relationships and representations underlying the nature of mathematics. In this we see that mathematics transcends physics and can tells us about aspects of the Universe beyond physics.
It is through mathematics that we trace back the unobserved expansion of the 15 billion year old Universe and discover that it has a finite age and a distinct beginning (a “naked singularity” unfortunately known as the “Big Bang”). It is through mathematics that we can define the mechanism of this birth and unfolding down to 10-34 of a second after inception. It is through mathematics that we may know our destiny and our purpose.
Before we discuss the future and the fuller meaning of mathematics, we should take a quick look at one other aspect of the Universe which has mathematical elements – the primary forces. There are four of them…
• The gravitational force: a weak and universal attraction between matter.
• The electromagnetic force: stronger than gravity and weaker than the nuclear forces, but relatively long-ranged attraction or repulsion between matter carrying electrical charge.
• The weak nuclear force: a very weak and short ranged interaction only within atoms (causing radioactive decay).
• The strong nuclear force: a very strong but very short-ranged interaction only within the nucleus of atoms (responsible for holding the nuclei of atoms together). It is generally attractive but can be act repulsively in some circumstances.
The mathematics of these forces is complex and involved. Even descriptions of the “basics (leading to a “grand unification theory or “GUT”) reads like something out of a bad nightmare…
A GUT model basically consists of a gauge group which is a compact Lie group, a connection form for that Lie group, a Yang–Mills action for that connection given by an invariant symmetric bilinear form over its Lie algebra (which is specified by a coupling constant for each factor)… When considering the symplectic gauge groups, the unification is even more complete when the irreducible spinor representation contains a right-handed neutrino and thus the complete particle content of one generation of the extended standard model with neutrino masses. Although it's probably not possible to have weak bosons acting on chiral fermions, a quaternion representation of the fermions looks like this:
And the “punch line”… A further complication with quaternion representations of fermions is that there are two types of multiplication: left multiplication and right multiplication. This can be explained in very simple diagrams like this…
I’m sure you understand all this (perhaps even more than I do). So let us try to look beyond the complex methodology and get to the point. Mathematics has led us on a search for a “theory of everything” (“ToE”) a single algebraic representation of the Universe in its simplest state - an all-encompassing, coherent framework of physics that fully explains all physical aspects of the universe. If such a representation exists (and the majority of physicists believe that it does), then it will show beyond dispute that the Universe is fundamentally mathematical.
Even without a complete ToE, we may now be relatively certain that the Universe began as a UNITY (aka “singularity”) which held the entirety of mathematics within its structure well before the first atom appeared. Not only did this UNITY contain a complete language which remains beyond our understanding, it contained all the information necessary to produce the Universe. But here’s the beauty of it. If the information of the UNITY was merely a stream of data or even descriptions of complex relationships, we wouldn’t be here to think about it.
Loss of Unity in the Forces of Nature
Time Since Beginning
All 4 forces unified
Gravity, Strong, Electromagnetic, Weak
Gravity separates (Planck Scale)
Strong, Electromagnetic, Weak
Strong force separates (GUTs Scale)
Split of weak and electromagnetic forces
The UNITY included a precise balance of deterministic and indeterministic aspects such that the major and long-term results could be predicted but never fully. (Picture the problems with either having everything predictable or having things too unpredictable).There is a very good parallel in biological DNA and cellular reproduction – although your DNA contains all the information needed to produce you, it would be impossible to predict all your physical qualities even with a complete copy of your genome. The best we could say is that some things are highly likely. It is perhaps this least expected balance which best indicates the apparent “intelligence” within the UNITY.
That is the ultimate nature of mathematics – it serves as a window to the Creation of the Universe. It reveals that there is an inexplicable and unexpected order underlying both the process and the result. It is far too complex and rational to be deemed random or inadvertent. To suggest that this beautiful and logical and profound system just merely “exists” (as if by magic) would be like entering a deep unexplored cave, finding an Oxford Dictionary and the complete works of Shakespeare, and then proposing that they just happened to emerge from natural events.
As Stephen Hawking has said: "The laws of science, as we know them at present, contain many fundamental numbers [constants], like the size of the electric charge of the electron and the ratio of the masses of the proton and the electron. ... The remarkable fact is that the values of these numbers seem to have been very finely adjusted to make possible the development of life." [Emph. Add.] Professor Hawking doesn’t attribute this “fine-tuning” to a creative entity (such as “God”) and instead suggests that the Universe invokes some process akin to “natural selection” so that only those values and processes that led to the present conditions are retained or utilized. Of course, this doesn’t address the existence of mathematics or the origin of those selection processes.
The “Fine-tuning” theories (including the anthropomorphic principle) lead to the conclusion that the Universe is fine-tuned for the building blocks and environments that life requires. There is no reason to re-state or add to the debate whether the Universe was “fine-tuned” to produce sentient beings (supposedly like us). The numbers speak for themselves and the debate never considers the existence of mathematics as an indication of an underlying “intelligence”. Scientists commonly reject the requirement of a Creator-God and religious folks tend to equate the evidence for a creative entity as inferring the existence of their god.
The point here is neither theological nor philosophical. We should follow the facts wherever they lead us. The facts that we “know” strongly indicate an underlying intelligence and purposeful design. Are there other explanations – perhaps. But all other theories require us to forsake the laws of causation (i.e. cause and effect) and seem to lack “common sense”. Those who argue that the theories proposing an underlying intelligence are mere tautology almost always use tautological arguments in opposition. And, to the best of my knowledge, none of the counter-arguments consider the place and existence of mathematics.
Oddly, one of the better considerations in the matter is brought from within mathematics itself. It is based upon a complex idea which is well accepted as proven: any formal theory expressive enough for elementary arithmetical facts to be expressed and strong enough for them to be proved is either inconsistent (both a statement and its denial can be derived from its axioms) or incomplete, in the sense that there is a true statement that can't be derived in the formal theory. This is Gödel's Incompleteness Theorem and its logic indicates that our attempt to construct a ToE is destined to fail. As explained by Stanley L. Jaki, because any theory of everything will certainly be a consistent non-trivial mathematical theory, it must be incomplete and this dooms any certainty in a deterministic theory of everything.
One might view this limitation as a notation built into the fabric of the Universe that informs us that even mathematics will not tell us everything. But then, this was obvious anyway.
To the best of our knowledge, mathematics is intended as a language in the physical realm. But it is not physical. It has no mass, exhibits no physical properties, and has nothing we can measure. Unlike the information which underlies the emergent properties of physical objects, mathematics exists as a metaphysical (beyond physical; not mystical or magical) attribute of the Universe. Like all metaphysical aspects of the Universe it remains largely misunderstood or not understood. We work with math, but we have little understanding of its true nature or origin.
Viewing mathematics as an intrinsic language of the physical Universe is a step towards better understanding its nature, but a deeper understanding will require exploration of its non-physical essence. Languages reflect the consciousness of their creator(s) – they are fundamental to thinking itself (try to have a thought that doesn’t invoke language). The language of mathematics reflects the consciousness which created it and that consciousness is largely beyond our grasp. So, like some of the math above, this can be a difficult and complex subject. We will try to reduce it to a few key concepts…
I will begin with symmetry. We generally conceive symmetry as axial invariance (a sphere appears the same when rotated through its axis). We are also familiar with mirror symmetry where an object appears symmetrical (agreeing in size, shape, and arrangement) in reflection. In common language, symmetry also refers to a sense of harmonious and beautiful proportion and balance. Physics and math use the term somewhat more rigidly to identify an inherent physical or mathematical feature of an object or system that remains unchanged under some transformation. Symmetries are very useful in mathematical formulations and can be exploited to simplify many problems. Why? Where does this transformational symmetry come from?
Like so many things at the core of physical reality, symmetry has no known cause. But without it, neither we nor our Universe would exist. Like other aspects with unknown origins, some argue that symmetry “just happens” and there is no need for a causative source. But symmetry doesn’t just happen randomly. Go to a beach or river bank where lots of rocks have been rolled and tumbled for very long times and try to find a round rock. No – not a rounded rock – a rock round enough or symmetrical enough that you could play billiards with it. If you find one, you’d be the first. Sure there are some round rocks, but never truly round rocks. Symmetry arises from specific processes intrinsic to the Universe.
But it is not the symmetry that we will focus upon here – it is “symmetry breaking”. The Universe seems to favor an initial state of symmetry in the quantum realm until it reaches some critical point where an infinitesimally small fluctuation breaks the symmetry and decides that symmetrical system's fate. Usually this involves a bifurcation or binary “choice” and the broken symmetry determines which branch is taken or asymmetrical state is selected. These “choices” appear arbitrary but these transitions almost always yield the same result: the system changes from a symmetric but disorderly state into one or more definite ordered states. Thus, most simple phase transformations of matter (gas, liquid, solid) and phase transitions (.e.g. crystals, magnets, and superconductors) are understood as spontaneous symmetry breaking. So what?
In 1964, physicist John Stewart Bell postulated that no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. Yeh, sure. Bell backed up his idea with a derived mathematical inequality that can be written as:
(ρ =the correlation between measurements of the spins of the pair of particles and a, b and c refer to three arbitrary settings of the two analysers).
Henry Stapp, a particle physicist at Lawrence Berkeley, has said: "Bell's theorem is the most profound discovery of science." Bell’s theorem shows that the nature of the Universe violates most general assumptions behind classical physics (not just details of particular models or theories). It holds that either quantum mechanics or Einstein's quasi-classical assumptions regarding locality and realism must be incorrect. Since experiments bear out Bell’s theorem and quantum mechanics, we must put aside our notions that the Universe prohibits superluminal action and that everything has a certain “reality”. In exchange for these losses, we gain “freedom” (indeterminism) at the quantum level.
But wait! There is another related mathematical revelation that changes our view of reality...
An argument that started three centuries ago about the very nature of objects in the Universe has led to one of the most striking revelations of this century – that quantum objects have a dual nature which is not resolved until someone or something “observes” their nature. Huh? Newton argued that atomic particles should be viewed as, well, particles. That seems obvious, but there was another group (led by Christiaan Huygens) who argued that atomic particles should or must be viewed as wave actions. For almost two centuries, the issue was debated – until a guy named Einstein came along.
This simple device was used to demonstrate that light (photons) are particles with mass as they strike the reflective side of the vane causing it to spin.
In 1905, before he theorized relativity, Einstein attempted to resolve a fundamental problem of the particle duality debate involving the “photoelectric effect”. Einstein postulating that electrons can receive energy from electromagnetic fields only in discrete portions (or “quanta” called “photons”). He found that the amount of energy (“E”) “captured” by a photon is related to the frequency (“f”) of the light by Planck's constant (Planck discovered this constant of angular momentum in 1900 and coined the term "quantum". His contributions to physics were magnificent).
This realization of a “simple” relationship between frequency and energy is probably more significant than E=mc2. Einstein’s work was expanded by Louis-Victor de Broglie who hypothesized that all matter has a wave-like nature. His idea was premised upon this equation…
which is a generalization of Einstein’s (since the momentum of a photon is given by p = and the wavelength (in a vacuum) by λ = ,). But then, along came Werner Heisenberg and his uncertainty principle (mentioned above). This principle states:
o x and p are the standard deviation (a measure of uncertainty) of a particle's position and linear momentum respectively.
o is the “reduced” Planck's constant (Planck's constant divided by 2 ).
Heisenberg explained his principle as an inherent consequence of measuring since any effort to measure a quantum particle’s position accurately would disturb its momentum and vice-versa.
And then along came John Archibald Wheeler in 1978. He came up with a strange idea on how to test photon behavior in light (PI) of the startling insight that everything in the quantum world has an indeterminate dual character. His idea centered upon another strange observation of waves called “interference”.Thomas Young first demonstrated this phenomenon in 1803 and his double-slit experiment was used to support those who advocated the wave nature of matter. Essentially, Young relied upon calculations relative to optics to support his claims. Anyway, Wheeler proposed a more modern application of the principle…
A plain light source (left) produces photons (light) which pass through one of two slits separated by a distance determined by the wavelength of the light. Light passing through the slits then diffuses and encounters light from the other slit. Because they are “waves”, the peaks and valleys at different intervals boost or cancel their energy – interfering with each other and producing the pattern of bright and darks areas on the right (an “interference pattern”). This effect is demonstrable mathematically from this geometric formula (Where the sinc function is defined as sinc(x) = sin(x)/(x) for x ≠ 0, and sinc(0) = 1)…
Wheeler proposed an experiment that has since been refined and improved such that specialized devices called “interferometers” may determine the state of a photon before it passes through a slit and afterwards. The results are astounding…
Once an observer begins to watch the particles going through the openings, the REALITY changes in accordance with the nature of the particle being tested. Since a particle can be seen going through one opening, it's clear it didn't go through the other and when under observation electrons or photons may be "forced" to behave like particles or like waves. The mere act of observation affects the resulting nature of the particle.
Richard Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment. Neils Bohr said: “Anyone who is not shocked by quantum theory has not understood it.”
The immediate conclusion is that one’s consciousness affects the behavior or nature of subatomic particles and since those particles make up physical reality, we consciously create or change reality through intention.
Now, we should note that this is a difficult “pill to swallow” for some and other explanations have been offered…
1. Particles move backwards as well as forwards in time and appear in all possible places at once, or
2. The universe is splitting, every Planck-time (10 -43 seconds) into billions of parallel universes, or
3. The universe is interconnected with faster-than-light transfers of information.
Of course, each of these ideas yields seriously problematic results and none have significant evidence to support them. Conversely, the Wheeler concept has been confirmed by numerous experiments using a variety of mechanisms and numerous particle types (including “buckyballs”).
So let’s pull all this together and conclude…
There is this remarkable, unexpected, and unexplained phenomenon or attribute of the Universe whereby everything we observe (physical things) has a corresponding logic and language which parallels and underlies its reality. We call this mathematics and tend to think of it as something we either create or originally discover. Instead, it is certain that it was there before us and transcends both our abilities and our understanding. While we think of ourselves (our brains) as the most ordered and complex things in the Universe, mathematics is far more ordered and complex than we are.
We have no idea where this mysterious math and its “magic” comes from. It is silly to simply say that it is inherent in the system or that it “just is” – especially in light of the dynamic and evolving nature of the Universe. To the best of our knowledge, nothing in the Universe “just is”. But, the most reasonable and easily supported answer is that mathematics was intended to be part of the intended Universe. More so, it is difficult to NOT see mathematics as providing a means of understanding both the processes of Creation and some aspects regarding the nature the Creator. Whatever else we might think we know about both the Creation and the Creator, we can say with greater certainty that they are rely heavily upon mathematics.
Viewing mathematics as the systematic symbolic representation of reality (which it sure seems to be), then we may also view it as providing a “solution” for the Universe. Not a Theory of Everything, but a real answer to the problems of how and why. The very existence of mathematics is compelling proof of an underlying intelligence in the design of reality. Nothing so complex, so powerful, so beautiful, so logical, or so predictive can be “random” or accidental. If we want to pray, we should do it with math (only slightly kidding).
We rarely stop to ponder why things “make sense”. Even scientists who use mathematics to discover new ideas about how things work or how things are related are neglectful in pondering the nature of the tool they use most in their work. By what accident is there a system by which we can figure out new relationships using a highly structured language that we didn’t create. And this language is the place we go when we wish to confirm that our ideas about reality are correct. Ideas about the physical Universe that don’t fit within the language of mathematics NEVER work out (or we realize that we don’t yet know the language well enough to make it work).
Mathematics is the key to understanding the Universe and we have neglected to ask where the key came from, why it fits the lock, and what other doors it might open. Those answers may have to wait until we figure out that mathematics also explains non-physical reality (such as consciousness).
RVW, December, 2014
A cross section of a quintic Calabi–Yau manifold A 2D slice of the 6D Calabi-Yau quintic manifold.
Some Other Useful/Used Resources and Notes:
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds (note above). Aren’t they beautiful?
"Dirac's Cosmology and Mach's Principle" by Robert H. Dicke, Nature 192 (4801)(1961): 440–441 - certain forces in physics, such as gravity and electromagnetism, must be perfectly fine-tuned for life to exist anywhere in the Universe. Dicke's considerations of a flat-space representation of general relativity with a variable speed of light led to led to a new theory of gravitation. His contributions to the study of the cosmic microwave background were substantial.
“Cosmic Coincidences: Dark Matter, Mankind, and Anthropic Cosmology” by John Gribbin and Martin Rees , Bantam New Age (1989). I also recommend “Just Six Numbers: The Deep Forces That Shape The Universe” by Martin Rees, Basic Books; First American Edition (2001).
The Anthropic Cosmological Principle by John D. Barrow and Frank J. Tipler, Oxford University Press (1898).
“Evolution from Space: A Theory of Cosmic Creationism” by Fred Hoyle and Chandra Wickramasinghe, Simon and Schuster, 1981 (Touchstone; Reprint 1984) - “Life cannot have had a random beginning...” “Once we see, however, that the probability of life originating at random is so utterly miniscule as to make it absurd, it becomes sensible to think that the favorable properties of physics on which life depends are in every respect deliberate ... . It is therefore almost inevitable that our own measure of intelligence must reflect ... higher intelligences ... even to the limit of God ... such a theory is so obvious that one wonders why it is not widely accepted as being self-evident. The reasons are psychological rather than scientific.”
“Encyclopedia of Physics” (2nd Ed.), edited by R.G. Lerner & G.L. Trigg, VHC publishers (1991), p. 495.
“Encyclopedia of Mathematical Physics”, Editors-in-Chief: Jean-Pierre Françoise, Gregory L. Naber, and Tsou Sheung Tsun, Academic Press (2006 ) (3500 pages).
A common sense interpretation of the facts suggests that a superintendent has monkeyed with the physics, as well as chemistry and biology, and that there are no blind forces worth speaking about in nature. I do not believe that any physicist who examined the evidence could fail to draw the inference that the laws of nuclear physics have been deliberately designed with regard to the consequences they produce within stars. Fred Hoyle
The forces that determined the expansion of the universe and seemingly produced a remarkably uniform universe is expanding away from an initial state of infinite density. The theories we have for the expansion can only approach this point. “As we go back in time, the conditions of the universe go beyond the limits of our knowledge; we can talk about a bang, but we can only study its aftermath.”
 Only after being impressed by Bertrand Russell’s “Why I’m Not a Christian” (http://users.drew.edu/~jlenz/whynot.html) and exploring his other writings did I discover that math was a MUCH bigger thing.
 Real numbers correspond to a point on the number line (as approximations). Rational numbers may be represented as decimals that have a finite number of digits or where the digits repeat themselves.
 We won’t get into the “collapse of infinities” or more depth about numbers.. How much is ? Oddly, it could either be 1 or
 A widely attributed, but poorly documented quote; most like from comments made during “Geometry and Experience", an address by Albert Einstein to the Prussian Academy of Sciences in Berlin on January 27th, 1921.
 π is an irrational number (it cannot be written as the ratio of two integers) and a transcendental number (it is not algebraic - the solution of any non-constant polynomial with rational coefficients). So what? Read on…
 There is nothing imaginary about this number – it is merely a new way of representing a real thing.
 A logarithm represents how many of one number we must multiply to get another number? For example, how many times must we multiply 2 to reach eight? 2x2x2=8, so the answer is 3. When derived in base 10 (decimal system), the logarithm is deemed “common”. The “natural log” is based upon the base e (Euler's Number).
 If you’ve never seen or heard of a slide rule, then you’ll want to Google it.
 Note the importance in calculus since loge x is the unique nontrivial function which is its own derivative and the function f(x) = ex (the “exponential function”) is the inverse of the natural logarithm.
 Taking the square root of less than nothing?
 Transcendental = abstract, supernatural, or metaphysical; being beyond ordinary or common experience.
 The motion equation in the transformed system showing the geometric and dynamic properties of spacetime.- the essence of General Relativity. See http://www.thphys.uni-heidelberg.de/~amendola/teaching/introduction.pdf
 "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik" by Werner Heisenberg, Zeitschrift für Physik (1927), pp. 43; 172–198. Note the relation to the Schrödinger equation. See http://www.math.ucla.edu/~tao/preprints/schrodinger.pdf.
 “A theoretical principle deduced from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present." "Law of Nature", Oxford English Dictionary (3rd ed.). Oxford University Press (2005).
 The integral of the Lagrangian of the physical system between two times t1 and t2. Luckily, we are not trying to explain this here – just noting the ability to represent this physical structuring with math.
 For those already knowledgeable about physical laws, you know that I am skipping many and just glazing the surface. For example, a proper discussion of thermodynamics also requires elaboration of thermodynamic equations. Equations of state describe the state of matter within a given set of physical conditions. They are constitutive equations which show the mathematical relationship between two or more state functions (such as its temperature, pressure, volume, or internal energy) of that matter.
 James Clerk Maxwell was one of the greatest thinkers in human history. Others have made critical representations beyond Maxwell’s such as Lorentz’s force law (the equation of motion for charges in the fields). See http://www.maxwells-equations.com/.
 Maxwell's equations are partial differential equations that relate the electric and magnetic fields to each other and to the electric charges and currents. As with all differential equations, specific boundary conditions and initial conditions are necessary for a unique solution.
 “QED: The Strange Theory of Light and Matter” by Richard P. Feynman, Princeton Univ. Press (1985), p. 129.
 A set of units is selected for dimensional physical constants so that their value is normalized to 1 solely as a convenience. The basis set may consist of time, length, mass, charge, and temperature or one might select a system of natural units defined such that the values of the speed of light, the universal gravitational constant, and other constants are all set to 1.
 aka the coupling constant for the electromagnetic interaction.
 While some suggest that pi is merely a mathematical constant, such is rather absurd – not only does pi represent something physical and measurable, it is inherent within several physical constants.
 The naïve idea that the Universe is expanding into some pre-existing “vacuum” remains widely held despite clear evidence to the contrary. Physicists cannot speak to what lies beyond or behind the Universe because “physics” simply doesn’t work there.
 “A Brief History of Time” by Stephen Hawking, Bantam Books (1988), p. 125.
 See "Populating the Landscape: A Top Down Approach" by Stephen Hawking & Thomas Hertog, Phys. Rev. D73 (February 2006), (12): 123527.
 See "How Bio-friendly is the Universe" by Paul Davies, Int.J. Astrobiology 2 (2003),(115): 115. I highly recommend “The Mind of God: The Scientific Basis for a Rational World” by Paul Davies, Simon & Schuster (1993). Another interesting read is “Intelligent Universe” by Fred Hoyle, Holt, Rinehart and Winston (1983).
 The computed probability of random events creating the specific conditions which COULD yield a viable universe is something less than 101000 to one (because we don’t know how many variations might yield viability). See “The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics” by Roger Penrose, Oxford University Press (1989). The odds are better of you winning the lottery every day for the rest of your life.
 Bell re-stated the theory in 1976 to corrected and expanded it. We should be sure that we are looking at this version.
 "Bell's Theorem and World Process", Henry P. Stapp, Nuovo Cimento, 29B (1975)(2), pp. 270-71.
 Einstein’s principle of relativistic causality rejects non-locality holding that that any causal influence cannot propagate faster than light (a causative event can only affect another event if light or energy could have travelled between them). “No combination of local deterministic and local random variables can reproduce the phenomena predicted by quantum mechanics and repeatedly observed in experiments.”
 See http://www.nature.com/news/physics-bell-s-theorem-still-reverberates-1.15435. Bell’s theorem has led to several misconceptions including the idea of an “observer based universe” and “multiverses” (as a way of avoiding or addressing the theorem’s conclusions).
 Young was perhaps the first to define the term "energy" in the modern sense. See http://en.wikipedia.org/wiki/Thomas_Young_(scientist) .
 The mathematics of double-slit interference in the context of quantum mechanics is based upon Englert–Greenberger duality relation. See "Quantum Optical Tests of Complementarity" by Berthold-Georg Englert & Marlan O. Scully, and Herbert Walther, Nature 351 (6322) (1991): 111–116.
 The very presence of the detector-"observer" near one of the openings changes the interference pattern of the particle’s waves as it passes through the slit. If the detector measures the particle as a “particle” instead of a “wave”, its nature changes to become whichever the “observer” is testing for.
 “The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory” by Brian Greene, W.W. Norton (1999), pp. 97–109.
 From “Meeting the Universe Halfway” by Karen Michelle Barad, Duke University Press (2007), p. 254 citing “The Philosophical Writings of Niels Bohr, Vol. 4, Causality and Complementarity”, Ed. by Jan Faye and Henry J. Folse, Ox Bow Press (1998), p.??.
 In a study reported in Nature (February 26, 1998, Vol. 391, pp. 871-874), researchers at the Weizmann Institute of Science conducted a highly controlled experiment which demonstrated that a beam of electrons is affected by the act of being observed, thus revealing that the greater the amount of "watching," the greater the observer's influence on what actually takes place. Hmm – I’m watching you!
 See "Diffraction of Complex Molecules by Structures Made of Light" by Olaf Nairz, Björn Brezger, Markus Arndt, and Anton Zeilinger, Phys. Rev. Lett. Vol. 87, #16 (2001), pp. 160401-1-4; “Quantum Interference Experiments with Large Molecules” by Olaf Nairz, Markus Arndt, and Anton Zeilinger, American Journal of Physics (2003), pp. 319-325. And, http://physicsworld.com/cws/article/news/2007/feb/15/photons-denied-a-glimpse-of-their-observer.
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