Fractals, such as this stack of spheres created using 3-D modeling software, are one of the mathematical structures that were invented for abstract reasons yet manage to capture reality.
Image: Illustration by Tom Beddard
In Brief
- The deepest mysteries are often the things we take for granted. Most people never think twice about the fact that scientists use mathematics to describe and explain the world. But why should that be the case?
- Math concepts developed for purely abstract reasons turn out to explain real phenomena. Their utility, as physicist Eugene Wigner once wrote, “is a wonderful gift which we neither understand nor deserve.”
- Part of the puzzle is the question of whether mathematics is an invention (a creation of the human mind) or a discovery (something that exists independently of us). The author suggests it is both.
More In This Article
Most of us take it for granted that math works—that scientists can devise formulas to describe subatomic events or that engineers can calculate paths for spacecraft. We accept the view, initially espoused by Galileo, that mathematics is the language of science and expect that its grammar explains experimental results and even predicts novel phenomena. The power of mathematics, though, is nothing short of astonishing. Consider, for example, Scottish physicist James Clerk Maxwell’s famed equations: not only do these four expressions summarize all that was known of electromagnetism in the 1860s, they also anticipated the existence of radio waves two decades before German physicist Heinrich Hertz detected them. Very few languages are as effective, able to articulate volumes’ worth of material so succinctly and with such precision. Albert Einstein pondered, “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?”
As a working theoretical astrophysicist, I encounter the seemingly “unreasonable effectiveness of mathematics,” as Nobel laureate physicist Eugene Wigner called it in 1960, in every step of my job. Whether I am struggling to understand which progenitor systems produce the stellar explosions known as type Ia supernovae or calculating the fate of Earth when our sun ultimately becomes a red giant, the tools I use and the models I develop are mathematical. The uncanny way that math captures the natural world has fascinated me throughout my career, and about 10 years ago I resolved to look into the issue more deeply.
This article was originally published with the title Why Math Works.
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62 Comments
Add CommentWith all due respect, I believe Mario Livio is missing the point about Mathematics. In my opinion, Mathematics is entirely a creation of the human mind and it works so well to describe Nature because the human mind evolved as a survival strategy for coping in that natural world: when in New York, act like a New Yorker. Through this coping strategy over billions of years, the human brain evolved an architecture which I believe mimics the dynamics in the world through its internal neural dynamics. Given sufficient neural complexity, a mind then emerges that is capable of self-awareness: it understands the world because its intrinsic dynamics are "like" the underlying dynamics in Nature. And from this mind, emerges a property we call Mathematics. Mathematics, the human brain, and Nature are all very similar dynamically in my opinion and that's why math works so well at describing Nature: Biology on earth chose as one of it's survival strategies, mimicry of natural dynamics through neural forms, and with sufficient time and complexity, a mind with self-awareness arose from this mimicry. Then from that awareness, further mimicry followed in the form of Mathematics. The human brain, Nature, and Mathematics are all very similar dynamically. It is this similarity in my opinion that gives rise to mind and the enormous success math has at describing Nature. There is no mathematics without the human mind: gravity controlled the orbits of planets before humans but there was no law of gravity before them.
Reply | Report Abuse | Link to thisFor survival one doesn't need to go beyond the everyday physical environment although it usually helps, no doubts. And mathematics goes way out beyond it... How could gravity have controlled the orbits of planets if no law of gravity was out there? That is because the so called physical reality is in fact ... mathematical by nature. For murky reasons some mathematical structures have apparently acquired the virtue of objective, "material" existence. If one stops and thinks about it, all physics become mathematical abstractions. Will all would-be imaginary mathematical structures one day emerge as objective in the empirical/physical sense? That's a million dollar question :)
Reply | Report Abuse | Link to thisIn order to prove that something was “created” it is simply necessary to prove that it did not exist before its creation. Therefore if it can be proven that something in Nature obeyed Mathematical principles before Humans used Mathematics then the argument is solved. Unless of course, we assume that Mathematics has an identical Twin that Mimics its effects in Nature so well as to be indistinguishable from Mathematical influence. So in the absence of proof of the existence of such a Twin and since I think we all agree that Natural forces worked the same before Humans walked upright as they do now, then we have our answer.
Reply | Report Abuse | Link to thisThe thing about creations and conventions is that they typically could have been done differently. For instance, cars may have had three wheels or Americans could have driven on the left instead of the right. But mathematics doesn't seem like it could have been done differently. There is no rethinking or improving 2+2=4. This is just how it is and it doesn't seem as if anyone could do anything about it even if they wanted.
Reply | Report Abuse | Link to thisThe only thing we could do is redefine the word 'two' to reference what is currently referenced by the word 'three' making the above equation false. Unfortunately, that is changing language, not math.
Galileo did not say mathematics is the language of science but of creation ,which is not the same.By creation he meant the world, the nature,the reality .Science is our imagination about that.
Reply | Report Abuse | Link to thisIn Galileo's time mathematics was a tool e.g. for calculate the strength of a bridge.Actually mathematics are inutile urnaments.
#1 Let’s assume a universe with a mixture of random and non-random processes.
Reply | Report Abuse | Link to this#2 Let’s assume organisms evolve in this universe that find survival value in“sensing” (make symbolic patterns from) selected repetitive processes in this universe. There is survival value in the repetitive sensations because the sensations call tell the organism what to do next. There would be no survival value in sensing random patterns.
#3 Let’s assume more complex organisms evolve in this universe that find survival value in creating a symbolic model of their universe of sensations with which to make predictions of “what will happen next.” (Animals) Of course, the models are only of value for the predictable aspects of the assumed universe so any data which conflicts with the symbolic model is assumed to be due to “random” processes and so is ignored when making these predictions.
#4 Let’s assume a “special” animal evolves which can make a symbolic model of this modeling processes. The animal calls this process “Mathematics”
With all these assumptions in place let’s ask:
What will the outcome of the animals use of his model of his own thought processes to make models of the assumed universe?
And: Why would it be surprising that this mathematically created model of the repetitive aspects of the universe agreed well with the observable repetitive portions of the assumed universe? - even to the point of making predictions that the animal’s own internal mind had not yet thought of?
Looks like a circularity to me. In other words, humans see what they have been designed by evolution to see. Hardly surprising. More interesting might be what they have NOT been designed by evolution to see. I mean what is really going on in what we regard as the “random” (non-mathematically predictable) portions of the model? Of course, human arrogance being what it is, we humans cannot imagine there really is any such thing.
www.MindMadeReal.com
ok.. to the last post, Mathematics even have a section that predict random ability of a sequence. There was an article on it by Scientific American just a few weeks ago.
Reply | Report Abuse | Link to thisHowever, did you know that Humans and Oxygen breathing life on this planet needs a specific amount of Oxygen in the air we breath to keep us living? Lower than 16% and Higher than 29% and we die... Is that balance for the last few million years an accident? A Random occurance?
There is so many random factors that ensure our existance and continued survival that the odds on it happening by chance is so high, it nears infinity, it would be like winning the lottery a million times in your lifetime....
Mathematics is a small glimspe of how God created the Universe. Scientist can tell you within a millionith of a sec what happened after the Big Bang, but nobody can tell you what started it...
I recognize the fundamental order and import of mathematics. I am, however, deeply saddened by the fact that my brain has never been able to comprehend it enough to fully enjoy it outside of its resultant implications on the world I live in.
Reply | Report Abuse | Link to thismaybe you didn't see that I said NON-mathematically predictable. You have just demonstrated my point that you can't even conceive of what you are not designed to translate into the equivalent of mathematical expression.
Reply | Report Abuse | Link to thisWe invent mathematics just as we invent language and theories. What we discover is how mathematics works in the real world. We invent concepts and symbols. We discover real world phenomena.
Reply | Report Abuse | Link to thisTheoretical physics is mathematics and invention. Experimental physics is observation and discovery.
It seems to me that the conservation of mass and momentum represent the axioms of the formal system that describes the universe. From my perspective if it weren't for the conservation laws, mathematics would be incapable of making any predictions. If middle world behaved like quantum world we would be surrounded by apparently non-caused events. But because of conservation, mathematics has an equal sign.
Reply | Report Abuse | Link to this"However, did you know that Humans and Oxygen breathing life on this planet needs a specific amount of Oxygen in the air we breath to keep us living? Lower than 16% and Higher than 29% and we die... Is that balance for the last few million years an accident? A Random occurance?"
Reply | Report Abuse | Link to thisFirst, Natural Selection is <i>anything</i> but random. Mutations are selected based on their propensity to promote survival.
Secondly, would you expect creatures on this planet (or ANY planet) to develop a requirement for something that doesn't exist in their environment? For example, would you expect a plant to develop a need for more iron than a planet had in it's atmosphere? If a plant like this developed, that would be a negative selector, the plant would die, and that variation/mutation would not be passed on.
Get your head out of a fictional book and apply critical thinking, please.
@Brynn
Reply | Report Abuse | Link to this<i>"However, did you know that Humans and Oxygen breathing life on this planet needs a specific amount of Oxygen in the air we breath to keep us living? Lower than 16% and Higher than 29% and we die... Is that balance for the last few million years an accident? A Random occurance?"</i>
First, Natural Selection is <i>anything</i> but random. Mutations are selected based on their propensity to promote survival.
Secondly, would you expect creatures on this planet (or ANY planet) to develop a requirement for something that doesn't exist in their environment? For example, would you expect a plant to develop a need for more iron than a planet had in it's atmosphere? If a plant like this developed, that would be a negative selector, the plant would die, and that variation/mutation would not be passed on.
Get your head out of a fictional book and apply critical thinking, please.
Jellyfish and Math
Reply | Report Abuse | Link to this"But imagine if the intelligence in our world resided not with humankind but rather with a singular, isolated jellyfish, floating deep in the Pacific Ocean."
(Page two of this article)
Now, would that jellyfish eventually develop numbers and mathematics?
Alright, here is an idea that is totally out of this world, but bear with me...
Imagine a 'human body', dressed and well... But now imagine that you are going to peel off layer after layer of that body. Of course you start with the clothes first, then you proceed with the next layer: skin, then the next: tissue, then flesh, then more tissue, then bone, etc.
Well, imagine that you have been peeling layer after layer until you couldn't peel off anymore.
Now, what you are left with is something in the shape of a (although watered down a bit)...
Just a minute...
What you are left with is cerebro-spinal fluid (CSF), the stuff that has the form of and is inside...:
(1) the meninges around your brain (for now keep the shape of those meninges in mind - it is like a deflated collapsed and dimpled plastic ball),
(2) your cranial ventricles (also keep their shape in mind),
(3) a long tube inside your central spinal canal (OK, you are keeping that in mind as well) and
(4) in your nerves (some of that CFS actually seeps out into your nerves and thus has the shape of long thin slimy strands - so you are keeping those long strands in mind as well.
Now seeing all that together, what does it look like?
It very much looks like a... jelly fish, albeit that the jelly is very much watered down...
What if indeed jellyfish were one of the earliest free moving living creatures who figured out a way of existing forever - in fact, there is a species of jellyfish that potentially can (Turritopsis nutricula)!
What if over a billion of years or so, a certain kind of jellyfish propagated itself in such a way that its survival became secured by eventually surviving (and multiplying) through the human species as... CFS, and that that this special jellyfish species through the human species (and perhaps other animals species as well) is still around inside of us and eventually came to invent... mathematics?
I got this idea in 1998 and somehow it does not want to let go of itself :)
You stated:
Reply | Report Abuse | Link to this"However, did you know that Humans and Oxygen breathing life on this planet needs a specific amount of Oxygen in the air... Is that balance for the last few million years an accident? A Random occurance?"
As I understand, oxygen was not a common element on Earth before it was produced over a period of several billion years as a product of photosynthesis by cyanobacteria. Perhaps it is life's intention is to opportunistically develop to survive on available energy sources...
I have to vote with the side that says mathematics was discovered rather than invented. It seems to me that if a particular set of math formulae that describes a particular physical property was invented, then one could argue that the physical property did not exist before some mathematician or scientist invented the math. Then the physical property sprang into existence.
Reply | Report Abuse | Link to thisFor more information about fractals, see www.fractal.org
Reply | Report Abuse | Link to thisMath is not invented and not discovered. It is part of logic that is inborn by humans. Some kind of logic is innate by animals also.The inborn logic is different for every human.There are "Born mathematicians" and probably Mario Livio belongs not to that kind of humans.
Reply | Report Abuse | Link to thisI do not know how and when the world was created.But I am convinced a Big Bang never happened as Don Paul believes.
Reading the comments, I get the definite impression that some people seem to equate the Name with the Thing. To me, Nature has components and properties that simply are the way they are and that were so well before we ever thought of them. We, humans, have developed ways of dealing with these phenomena and the most succesful way of doing this, is thru mathematics.
Reply | Report Abuse | Link to thisOf course we invented it. This "notation" has proven itself to be a most powerful way of describing objects and their behaviour and, as such, it enables us to make predictions etc. But surely this doesn't mean that our description determines the objects or their behaviour.
So, is Nature mathematical? No, not to me. Mathematics is simply the most succesful way of dealing with many of its aspects, but this doesn't mean that - in principle - no other way could be found to do the same thing. Of course you could ask yourself if it would be worthwile to try to find such an alternative to Math.
I think there's actually a simple answer to the second question, about mathematics' explanatory and predictive power: computational (or Turing-) universality. Any computationally universal system is capable of emulating any other computationally universal system, and sufficiently sophisticated mathematical systems are computationally universal (the computational universality of number theory was really the crux of Gödel's proof of his incompleteness theorems). Nobody wonders why computers seem to be able to simulate all kinds of physical systems to arbitrarily great accuracy; but the reason is no different for mathematics.
Reply | Report Abuse | Link to thisIndeed, one datum we have to support this idea is the existence of ontologically inequivalent, but equally applicable models for physical systems -- think things like the AdS/CFT correspondence, where two theories exist that disagree on things as basic as the number of dimensions of spacetime, yet describe the same physics.
Mathematics describes the world in the same way computers simulate it -- through emulation, or imitation. The same way any universal Turing machine can emulate any other, mathematics can emulate the physical world (at least to the extent it is computable).
I've written a somewhat more in-depth post about this here: http://ratioc1nat0r.blogspot.com/2011/07/universal-universe-part-iii-answer-to.html
Wow. God did it. We could as well ask the Taliban. Same answer.
Reply | Report Abuse | Link to thisLet me guess. It was your god of course, not one of the thousands of others that dwell around. OK then, I go along. Your version of God, your version of the bible. But please tell us, in your gods own universe, before he decided a spin off, could 1+1 be 3? Could 1/2 make 100? Could Pi be 25? Could the formula for the Mandlebrot graph be different? Could a perfect square have only 3 sides? I mean, without cheap tricks and funny stuff we're all familiar with to amaze or fool friends. Seriously. I can tell you the answer: no. By definition no. Logic is not designed nor invented. Logic exists because it can't be any different. Not in an absolute void, not in the most magical reality. Yes, things can be different in our imagination and in our dreams, but we're talking about reality.
Another approach: an hypothetical intelligence cannot invent logic if there weren't logic yet. It could not even exist. Since there actually IS logic, mathematics if you like, this preceded your deity. Further, an intelligence would have to be somewhere and for some time and experience time to do something. Anything. That's already a quite complex framework, which also preceded your deity by definition. It's impossible to exist nowhere and never first, then design your own habitat to be able to design something, before even starting to design a universe and design multicellular organisms where some 'clever' species are constantly fooled into thinking the universe is much older than 6000 years.
Sure it is possible this universe was created by a lifeform in another universe. But a god by the very definition of a god? No. An alien at most. Bound to the very same laws of nature, logic, as we are. No magic things. Only technology. And thankfully there is absolutely nothing seen, measured, or even remotely detected that points in the existence of a super alien like that, let alone a 'god'.
'And those bronze age myths, what about them' you might ask. Well, those bronze age myths explain a kind of universe that does not match reality. Simple. We know that because our modern models give us means to make advanced technology. The kind of universe explained in those scriptures perfectly match the very limited world views of bronze age people. Sure, there are some universal truths in there like "don't kill" and "don't steal", but that's it. And it's even far from unique. Ask the Taliban.
Flapdoodle!! The brain evolved to survive, and to do so creates a model of the world that enables the organism to escape danger, pursue food and mates, and generally navigate the environment. The brain DID NOT evolve to apprehend the true nature of reality. (Investigate the neurology of vision, esp. the nature of "optical illusions" to convince yourself.) Mathematical relationships exist a priori and are discovered, although, of course, aspects of mathematics are a creation on the mind.
Reply | Report Abuse | Link to thisDutchguy, do you understand the concept of God? You made a real good case why Zeus doesn't exist. But these are simplistic arguments you offer that hardly stand up to close scrutiny.
Reply | Report Abuse | Link to thisYou seem to be unaware of the western concept of God in Judaism and Christianity. And completely unaware of the three centuries of philosophical arguments from the ancient Greeks through Descartes through Kant through modern philosophers. Before you take a firm position on the pre-existence of logic, at least read some of the centuries of philosophy and some of the more profound religious writings on the subject.
You don't seem to even understand the concept of God.
Typo in comment 24. Three centuries should read "three millenia."
Reply | Report Abuse | Link to thisHeavens exist.
Reply | Report Abuse | Link to thisNo it doesn't...
Reply | Report Abuse | Link to thisI tend to think that mathematics is discovered. If I say ``there are two prime numbers between 20 and 30'' (23 and 29), is that statement ``True'' before I made the sentence? Most people would say `Yes'. The hallmark of mathematical truth is that ``it couldn't be otherwise''.
Reply | Report Abuse | Link to thisWhat fascinates physicists is why the universe is describable by mathematical formulas. It is fascinating that the gravity should follow the (1/r^2)- law. Could it be otherwise? Why not (1/r^3)? Why not (1/r^2.45)? It is amazing.
But wait! I heard one physicist saying something to this effect: ``Isn't it amazing that 5+7 equals 12? Isn't it amazing that if we add 7 stones to 5 stones we will get 12 stones?'' Even this simple plain axiom is questioned. I'm only beginning to appreciate his statement. It couldn't be otherwise, could it?
Math is built on nature, man has DISCOVERED it and INVENTED symbols to to make it readable and understandable by humans.
Reply | Report Abuse | Link to thisI guess our knowledge of math is just a process of discovering the controlling rules of the universe's nature.
Reply | Report Abuse | Link to thisSo the sum of the inner angles of the square of the triangle "just for example" is not an invention at all, but it's the nature of the shape which we discover by the experiments.
But I think the answer would be more complicated to explain all the rules, but that was just my simple answer for now.
When writing an expression evaluator, I've found that the order of operations is in fact arbitrary. If you change the operator precedence, you of course, must also change the way you define expressions. This suggests that the order of operations is a "standard" allowing everyone to do math the same way. Such as, what side of the road you drive on... The operations themselves, for lack of a better term, exhibit natural mathematical properties of numbers.
Reply | Report Abuse | Link to thisThe evolution of our Universe must, by necessity either be random or ordered. If it is random this does not preclude the “appearance” of Order on some scale, but if it truly is ordered then any appearance of randomness is merely a reflection of our inability to discern the underlying order. So the question of Order vs Randomness may be insolvable by us at any given point in time for a given set of observations, but whether or not we give this Order a name and say that it is controls Universal Evolution through a set Laws that we are able to understand does not infer that any observations we may make of the intricate Nature of such Order confers on us the title of its “Inventor/Creator”. Clearly any “method of description” we attach to describe, and facilitate a clearer understanding of, the workings of the “Order” of the Universe should not be confused with the actual workings of the underlying Order.
Reply | Report Abuse | Link to thisThis discussion reminds me of a scene in one of Tom Hanks movies where he is on his knees, on a beach, before the fire he has just managed to start, raising his arms up to the sky, proclaiming “Look what I have Created”. Does anyone, other than he, actually believe that he has “invented/created” Fire?
So if we accept as apriori that the natural forces at work in the Universe existed prior to emergence of Humans as thinking Species then the argument for our creation of the underlying descriptive Language at work is settled. We, as Human Beings, understand and thus describe the Universe according to the limitations and abilities provided to us by our Genetic Architecture.
It may very well be that as we dig deeper into the Fabric of our Reality our current Mathematical tools may fail us and we will be forced to come up with something else to accurately Model that which we observe. I suppose then we will say, once again, “Look what we have created”.
Consider the following: How would another sentient Species describe the workings of our Universe? Would they understand its apparent order as a Logical progression of Mathematical driven Laws or would they perhaps “feel” its order as a harmonious balance of, say, musical-like notes whose patterns are perceived and visualized in their Alien Brains such that when the complete “Tune” is heard and in balance, so to speak, a level of perception of understanding is reached which satisfies/answers their questions surrounding the reality that they are experiencing?
Would they then be permitted to exclaim look at the “Tune” I have created?
I believe the answer to the question of why mathematics, which is structurally simple and was "invented" or "discovered" by the human mind, is at the same time capable of very accurately describing and predicting the natural world, can be found in the very place that scientists have been striving with all their might to deny or to explain their way around -- namely, the Bible and God. Now I am a scientist myself by upbringing, by training, and by conviction; and I've been an avid reader of Scientific American since my youth. Yet at the same time I am religious and fully believe in God, and that in God and the Bible can be found the fundamental explanations for many of our riddles and questions in science itself and in life in general. The first sentence in Genesis reads, "In the beginning, God created the heavens and the earth." A bit later it reads, "And God said, let there be light, and there was light." The same pattern proceeds for the entire creation process. This could have taken billions of years, of course, and was not simple. But the point is that there was a plan and a purpose in the beginning. If we stop denying that hypothesis, and look it straight in the face, we can immediately deduce plenty of logical implications which bear out in observation and experience. The mathematical nature of reality is one example. Thus, the explanation for why nature is apparently mathematical is that it was designed so by God, who has a logical mind, just as we do. So, if we would plan to do something, we would organize it on a logical basis in order to make it work, without undue complexity or vagaries of whim -- and this hypothetical "God" would do the same. So it is entirely reasonable that God thinks in a mathematical way, nature which he/she created is mathematically orderly, and likewise we humans have a mind which operates in a similar manner. In a sense, this also agrees with the first comment submitted, in that humans are optimally adjusted to the universe in which we live. The difference is the teleological philosophy of my argument versus the evolutionary philosophy of his approach. Still, the end result is the same.
Reply | Report Abuse | Link to thisi would believe that mathematics is an invention of man. If we take into account the uncertainty principle, you would see that the calculations we use to relate to the physical world become probability. while that ratio may be so minuscule to the chance of something varying away from the norm (for example, the moon suddenly leaves its orbit around the earth), it's still only a probability. We can only use math as our language to interpret and predict the functions and actions of the natural world around us. however without us to create these interpretations in mathematical form, math as a language would not exist.
Reply | Report Abuse | Link to this"When in New York, act like a New Yorker." Except math works in New York, Paris, Munich, and Mars. We may invent the symbols we use to communicate math, but not the relationships - the "why/how it works" part. If you decide "oxlib" means "to gather daisies", and enough of your friends buy into it - great! But if you decide 3 = 7 or a + b = a - b for all values of a and b, everyone in the world may agree with your for all time. But it's still wrong.
Reply | Report Abuse | Link to thisWhere in Bible did you see mathematics, WilliamStoertz?
Reply | Report Abuse | Link to thisYour picture looks mythological. Order emerges from Chaos/Randomness and dissipates back to latter. Completely deterministic dynamics could be chaotic, exponentially sensitive and unstable with respect to its current status.
Reply | Report Abuse | Link to thisYour powers of perception must be finely tuned since I was, at one time, an avid reader of Greek Mythology. Having said that, would you mind expanding on your statements so I can get a better handle on what you believe I believe.
Reply | Report Abuse | Link to thisI'm a bit surprised that many participants here don't seem to be aware of what science says. They speculate on their own but modern science has since long moved beyond triviality of everyday conceptions. Unfortunately I'm reluctant to expound further on because it would take too much effort and space. I would only reiterate that physical reality is woven out of mathematical abstractions, it's at least a proper part of mathematics. Will all mathematical structures and representations acquire one day objective physical existence we don't know.
Reply | Report Abuse | Link to thisWhile I certainly would not want to appear offended or, indeed offend anyone else in particular, I must say that, I don’t know about anybody else, but I’ am always suspect of the motives of anyone who tells me something to the effect that “it would take to much time and space” to explain themselves. While I don’t profess to know everything that Science tells us; I do know that it tells us that the observed passage of Time is relative to the Observer’s Frame of Reference and so by the use of this, now trivial everyday conception, I must conclude that apparently certain individuals may be moving so fast Relative to the rest of us that they believe that any attempt we may make at translating what they might construct in the Infinite Time available to them, may put us less-accelerated Mortals in danger of exhausting the apparent Finite Time available to us, apparently.
Reply | Report Abuse | Link to thisMaths in nature works by brute force. There is usually only one most efficient packing method which gravity uses when it can. Most packing methods are chaotic, (consider the shapes of asteroids, all governed by gravity). In school we are taught the formulae for regular objects, not the mess of chaotic numbers which determines the configuration of the universe. Pure maths is the intellectual idealism of those who deny the prevalance of turbulence.
Reply | Report Abuse | Link to thisIt's simpler than your last post alludes to :) . A fruitful discussion suggests, imho, sticking to the specific and precise notions of contemporary math and science and trying to interpret those. I would shy away from banking on vague terms like Order vs. Chaos/Randomness, Infinite Time vs. Finite Time etc. I certainly didn't want offending anybody but behind these vague terms there are rich and elaborated areas of mathematical work and scientific evidence that better be accounted for. For example, it occurred to me that you were rather favoring Order/Determinism vs. Chaos/Randomness yet the latter is represented at the most fundamental level by the empirically observed yet unpredictable localization/collapse of the non-local/entangled probability amplitudes in Quantum Mechanics.
Reply | Report Abuse | Link to thisWe are not technical/mathematical here, in these comments, but the only mythology which seems deserving to remain with science as even (!) an ideal is Hegel's Logic. The latter hints at a world which could not have been non-existent, different it could not have been either, there had not been and could not be another such one.
Reply | Report Abuse | Link to thisWe are not technical/mathematical here, in these comments, but the only mythology which seems deserving to remain with science as even (!) an ideal is Hegel's Logic. The latter hints at a world which could not have been non-existent, different it could not have been either, there had not been and could not be another such one.
Reply | Report Abuse | Link to thisthis is better: ... another one could have not been and could not be out there.
Reply | Report Abuse | Link to thisNo offense was taken here. I was merely attempting to appear “clever” by playing with words. In any event I previously asked you what you believed, I believed, because sometimes my translation of thought to written language leaves a lot to be desired. The simple fact is I do not know, and indeed would never presume to know, if what we call this Universe is Ordered or Random. The existence of the obvious relationship between logically proven and accepted Mathematical Laws and the workings of the Universe would seem to indicate the possibility of an underlying Order of sorts and since this article is focused on the relationship between Mathematics and its relationship to the workings of our Universe I was trying, apparently unsuccessfully, to point out that what may appear to be Random may in fact be ordered and vice versa. I personally feel that there is insufficient evidence to definitively state, at this time, that we know the correct answer to the question of Order vs Randomness. I will admit that given the observational data at present Randomness appears to be in the "lead", but this is why I stated that (“any appearance of randomness is simply a reflection of our inability to discern the underlying order”) I probably should have said “may simply be” instead of “is simply” and I should have added the caveat “at this point in time” to my statement.
Reply | Report Abuse | Link to thisWhenever I think about the possible make-up of the Universe I ask myself the following: Could it be possible to have at once an ordered Universe and a Random Universe existing at the same time? If I accept that the appearance of Randomness may exist within a much larger system of Order and that the appearance of Order may be simply be a consequence of the “sample size” taken from a truly Random Universe then I believe it may be possible.
Think of it this way We go about our daily lives in what we consider to be an ordered fashion having no regard for the Randomness that appears to permeate the Quantum level and up until recently we did not even know of the existence of the aforesaid random nature at the quantum level and we suffered no obvious deleterious consequences as a result of our ignorance, at least at the level of our daily lives.
I am not sure why, but Russell's Paradox keeps popping up in the back of my mind when I contemplate this question of Randomness vs Order in the Universe. I will have to give that some thought to see if it leads me anywhere.Usually when I think of that Paradox it leads me to headache,but oh well that is what aspirin is for.
Mathematics is predominantly random as well. Most problems with infinite input are undecidable by algorithm. That means one has to device ad hoc solutions for each individual case or at least for classes of such cases. Arithmetic is incomplete (Gödel) - that means infinite number of mutually independent and hence logically random to each other axioms are needed to account for all arithmetical notions. Most problems in combinatorics (Kepler's sphere packing, 4-color mapping etc.) are amenable only to brute force by computers because they simply don't possess any structured/ordered solution space.
Reply | Report Abuse | Link to thisRussell's Paradox was about the barbers who don't shave themselves, right? :) It never struck me as a real logical problem, just a linguistic curiosity. The same for so called paradoxes of early/"naive" theory of sets. Those were easily made disappear by axiomatics but the real logical problems remained: Cantor's Continuum Hypothesis , Zermelo's Axiom of Choice etc.
Actually the Barber’s Paradox example was concocted in an attempt to put a simpler to understand face on the original paradox postulated by Russell, which concerned Naive Set Theory. This is probably why that particular Paradox surfaced in my mind when I was thinking about the question of Order versus Randomness because what I actually was thinking of was the possible application of the logical constructs of Set theory and how they may be applied to this question. In any event I am still a long way from straightening this out so that it may be somewhat justifiable and understandable to myself let alone anyone else.
Reply | Report Abuse | Link to thisStrange amount of overkill going on here: and it's due to the misleading point of the article! Duh! Mathematics: is a tool for thinking about and analyzing natural systems which may have existed since the beginning of the Universe but are also "local." That is, humans encounter phenomena which come to our awareness through out senses. When a person is made aware that there's an avalanche bearing down on them, they know it because of their eyes and ears. "What to do" about the avalanche doesn't require the invention or use of mathematics. But we must understand the causes and nature of avalanches in order to develop systems to either prevent or avoid them.
Reply | Report Abuse | Link to thisSo, the brains get working and soon we have teams firing cannons at snow drifts high on mountain sides. We engineer warning sirens. We need math to make cannons, to know the trajectory of cannon shells. We need math to create electronic sensing and sirens.
In other words, mathematics is a creation enabling other kinds of creation. And: mathematical knowledge is accrued, expand and is passed on, so that later generations have a mathematical system in place to create more mathematical systems which don't just "describe" reality - they give us the means of responding to aspects of reality which might harm us.
Posing the pre-existence of mathematics prior to the arrival of humans is similar to the question about falling trees in a forest empty of humans - the falling trees make sounds even though no human hears them. Sound exists independent of human perception. Math: does not exist independent of humans, it isn't an agent producing anything until a human creates and uses it.
Mr.Livio raises a number of interesting points, however, it all boils down to a which came first the chicken or the egg question.
Reply | Report Abuse | Link to thisIn the real world of science there are constants and laws of nature and they are as described in the following quote as ‘just so’ and these constants and laws govern the universe and all that is in it.
Quote: “Why are the constants and laws of nature just so, and not different? For example, why is the speed of light not faster than it is? Why are electrons so much lighter than the protons they orbit in atoms? If fundamental laws and constants were even slightly different from what is observed, then life as we know it would not exist. (For example, atoms would be less stable, or stars and planets would not form.)” End of Quote. Source: Universe Forum, Harvard Smithsonian Center for Astrophysics.
In other words why are there constants and laws of nature, physics, that have governed the universe since its creation ? It seems that mathematics evolved so that humans were enabled to know, i.e. discover, that which existed from the beginning. What is truly amazing and marvelous is that through very rigorous and complex mathematics the laws of nature turned out to be simple equations, for example, Newton's laws of motion, Einstein's E=M x C (squared) amd the Mandelbrot set.
@ Milioto: I absolutely AGREE with you. It is unnerving that I found your posted comment a few hours after I came to that conclusion myself after reading a mathematical article in the May 2011 issue of Scientific American ("the strangest numbers in string theory")that I came across as I perused for the sake reading what I had missed when I first bought it. I kid you not when I say that it was akin to an epiphany the realization that our mental processes have developed analogous to the universe- beyond the mere scope of Nature. Just as astrophysicists have theorized that the universe has a set amount of elements and matter that were present during "the big bang" and has not and will not ever change, the human mind has a dynamic microcosm of this cosmological phenomena and characteristic. In a nutshell, it is advent and inevitable for humanity to create forms and distinct expressions of intelligible, practicle, and insightful purpose that can "translate" what the universe is "saying" all around and WITHIN us.
Reply | Report Abuse | Link to this@ Milioto: I absolutely AGREE with you. It is unnerving that I found your posted comment a few hours after I came to that conclusion myself after reading a mathematical article in the May 2011 issue of Scientific American ("the strangest numbers in string theory")that I came across as I perused for the sake reading what I had missed when I first bought it. I kid you not when I say that it was akin to an epiphany the realization that our mental processes have developed analogous to the universe- beyond the mere scope of Nature. Just as astrophysicists have theorized that the universe has a set amount of elements and matter that were present during "the big bang" and has not and will not ever change, the human mind has a dynamic microcosm of this cosmological phenomena and characteristic. In a nutshell, it is advent and inevitable for humanity to create forms and distinct expressions of intelligible, practicle, and insightful purpose that can "translate" what the universe is "saying" all around and WITHIN us.
Reply | Report Abuse | Link to this@Patrick49: The regulation of how things are are always argued in mathematics. That is one of the most amazing qualities of the faculty; that we can model, mimic, and undo real situations in mathematical form. In math we can experiment with the nature of the known and not yet known, safely.
Reply | Report Abuse | Link to thisMathematics is a powerful and impressive TOOL for mankind to generate a plethora of data and knowledge. Imagine if every single human being came to that revelation and fact, that we weild that ability and that we, as a whole, can harness countless more... therefore, we SHOULD value ourselves SO MUCH MORE than we presently do (being sentient beings capable of remarkable things)... LOL, the lack of mutual respect apparent through the ages is the evidence of spirituality... but that is too deep for most to comprehend.
Reply | Report Abuse | Link to thisIntelligentsia is a definition in a dictionary and not necessarily an avidly friendly state of being in regard to the temperaments of individuals and their corresponding reactions/ responses. "/
Reply | Report Abuse | Link to thisAs a simple old guy who could never really relate to mathematical enthusiasts, I have to point out that the first anecdote in this article describing how Maxwell's elegant mathematical description of what was known of electromagnetism but that "they also anticipated the existence of radio waves two decades before German physicist Heinrich Hertz detected them" did so because EM waves are physical manifestations of electromagnetism. It was not Maxwell's foresight that allows his description of electromagnetism to apply to radio waves.
Reply | Report Abuse | Link to thisAs for Einstein's expression of fascination with mathematics, "How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?", I doubt that the methods of decimal mathematics would have ever developed if we had had hooves instead of fingers and toes. I suspect that our system of mathematics may be a development of our physical experiences with our fingers and toes...
So many of the twentieth century mathematicians and scientists seem to think that mathemtics began in the 16th century.Major mathematical discoveries were made prior to the 1600's and along with these much work was done in the philosophy of mathematics. The author of the article referred only to Platonism and the form of nominalism which is called Formalism but did not even consider realism derived from Aristotle and the scholastics of the middle ages. In the realist epistemology a bond exists between the world and mathematical concepts. Mathematics originates by observation of extra-mental reality and by abstracting upon these observations the mind comes to the ideas of number, infinity and so on. In the nominalist or Formalist epistemology mathematics is entirely constructed by the mind and this is the reason that the author of the article has such trouble in finding a bond between the world and mathematics. From this point of view a unicorn has just as much "mathematical reality" as a point, line or derivative. Since extra-mental reality has been discarded there is no corrective mechanism in place to prohibit the mind from spurious mathematical reasoning.We can see this bourne out in the mathematical systems where even the principle of contradiction is denied.Such nominalist mathematical systems must be and remain inconsistent and incomplete because their axiomatic foundations are only products of the mind. When writing such articles in the future it would be prudent if the author included more mathematical schools to get broader perspective of the relationship of mathematics and reality.
Reply | Report Abuse | Link to thisThe writer concludes
Reply | Report Abuse | Link to this"There is no mathematics without the human mind:
gravity controlled the orbits of planets before
humans but there was no law of gravity before them."
My reading of this is that the writer is victim to the
irony that although he is ready to admit the existence
of gravity independent of it's study, for some obscure
reason mathematics isn't granted the same distinction.
May I suggest that it is due to an idiotic materialism
endemic in the material sciences.
JONIMETHFAN.
Correct, Stewart Halpern.
Reply | Report Abuse | Link to thisI second your statement:
Mathematical relationships exist a priori and are discovered, although, of course, aspects of mathematics are a creation on(of)the mind.
Well said.
I think both views are accurate. We INVENT the languages we use for mathematics, e.g. we mostly use base ten arithmetic because we have ten fingers. We DISCOVER theorems with proofs, which are eternal truths which were true before we even thought of them, e.g. prime numbers. We know there are an infinity of primes but we have a recreation of proving the latest largest known prime. How could we discover a new prime if primes did not exist seperately from our discovery?
Reply | Report Abuse | Link to thisEvolution only explains how our minds could be suited for Newtonian mechanics or game theory, not QM | K-theory | GR.
Reply | Report Abuse | Link to thisThat's right. Thurston put it well here: http://arxiv.org/abs/math.HO/9404236
Reply | Report Abuse | Link to thisWhy Math Works ?
Reply | Report Abuse | Link to thisBecause math tied with physics.
For example:
I say that there is circle-particle that can change /
transformed into sphere-particle and vice versa
and Euler’s equation cosx + isinx in = e^ix can explain
this transformation / fluctuation of quantum particle.
I try to understand more details.
I have circle- particle with two infinite numbers: (pi) and (e).
I say that this circle-particle that can change into sphere-particle
and vice versa. Then I need third number for these changes.
The third number, in my opinion, is infinite a=1/137
( the fine structure constant = the limited volume coefficient)
This coefficient (a=1/137) is the border between two
conditions of quantum particle. This coefficient (a=1/137) is
responsible for these changes. This coefficient (a=1/137) united
geometry with the physics ( e^2=ah*c)
=..
If physicists use string-particle (particle that has length but
hasn’t thickness -volume) to understand reality
(and have some basic problems to solve this task) then why don’t
use circle-particle for this aim.
It is a pity that I am not physicist or mathematician.
If I were mathematician or physicist I wouldn’t lost the chance
to test this hypothesis.
=..
Best wishes.
Israel Sadovnik Socratus
==…