“When we do mathematics, we touch immortality.”
quantamagazine.org
quantamagazine.org
But apart from that, I find this sort of talk unhelpful. It's a bit like Schopenhauer who thought that instrumental classical music gave access to the innermost truth of the universe (I'm paraphrasing from memory here) - what hogwash.
Lastly, the headline has not much to do with the article, has it?
How do you know it's hogwash?
If you're asking for a proof, I'd say that the fact that multiple equally sophisticated musical schema are present across the world in different cultures that he didn't acknowledge (and only _instrumental_ music??).
The point that I think you inadvertently bring up is that an appreciation of mathematics is so far only human (and we tend to focus on the ones that are useful to us right now)... perhaps when we meet alien life we'll actually know how universally interesting our little parts truly are. In any case, they are still mostly consistent sets of postulates, theorems, and axioms.
Basically I believe that instrumental music conveys no deeper truth (for starters, it hardly conveys anything without significant training on behalf of the listeners. Program music like Berlioz' Symphonie fantastique is called program music after all because Berlioz himself handed out a program, notes, explaining (in words) what was going on in the symphony (the artiste falls in love unhappily and poisons himself with opium and has nightmares). Now, you can immerse yourself in profound romantic feelings when listening to it, or find delight in the unexpected deviation from the rounded binary form in a minuet & trio, or "suffer" from a deceptive cadence, but that's because it's previously been communicated and learned, with words and examples. Plus it's not exactly "secrets of the universe" stuff.)
Wagner's Gesamtkunstwerk operas also include words, and to understand them, it helps to get an explanation and interpretation (with words).
Of course some people claim to have had deep insights under the influence of psychedelics, and some people claim to have had deep insights from classical music. What I'm basically saying is that the claim is not enough, you must also be able to communicate this deep insight.
And basically I claim the insight must be communicated in words. And then the words carry the insight. Not drugs or music. Visual arts or music (or drugs?) can support that communication, but they can't replace it.
Of course, you could just say, well, there is all this insight in the music, and Schopenhauer got it, and you get it, and I don't, and you cannot communicate it in words, you can just communicate it in music, and so it's there, but I don't get it. That's a bit like what postmodernists like to do when someone challenges the scholarship of their papers - "you just don't get it". And that I call hogwash, I suppose, for now.
It might be instructive to listen to an hour of Cantonese opera, and see what insights are gained.
[1] https://www.theguardian.com/books/2004/may/06/classics.georg...
Granted, a bit harsh :-) There's more to be said about it. But remember how Ford Prefect wrote volumes about earth in the Hitchhiker's Guide to the Galaxy, and then it was abridged to "Mostly harmless"? So that was (my) very abridged version of idealism.
Euclidean geometry is a human invention. In the real world, spacetime is curved.
Real numbers are a human invention. The universe does not seem to be built of numbers with infinite precision.
I would go further and say that the number "two" is a human invention. The universe could carry on just fine without any life anywhere having the concept of "two". But the case is certainly stronger for calculus and geometry being human inventions, than it is for number theory.
Why is curved spacetime (in the sense of an infinitely differentiable pseudo-Riemannian manifold with a Lorentzian metric) not a human invention while Euclidean geometry (a limiting case of such a manifold) is?
> The universe does not seem to be built of numbers with infinite precision
What's your definition of spacetime? It's not that of General Relativity, if you deny infinitesimals.
I think you're stuck either with accepting curvature and infinitesimals and the involvement in pi in the relationships between a set of points at a small fixed spacelike distance from an origin, or you're stuck with denying curvature as anything other than a coarse approximation of some manifold with a non-Lorentzian taxicab-like metric.
Minimum lengths (abolishing infinitesimals) have been proposed theoretically, and we can readily parameterize with a minimum length equations like the Dirac equation or the Klein-Gordon equation or like the Lagrangian of the Standard Model, and in doing so we see pretty plainly that the Lorentz transformations of Special Relativity must be modified. One concrete modification is the theory of Doubly Special Relativity whose length cutoff is the Planck length. It predicts an energy-dependent speed of light, and that is contradicted by evidence (gamma rays do not win races with infrared rays in near vacuum). Other similar deformations of Special Relativity have been advanced, and they also conflict with experiment and observation, unless the minimum length goes to zero, or other physical quantities are (counter-)parameterized pushing the observation-and-experiment conflict elsewhere. (One programme for this sort of research is the Standard Model Extension).
> The universe could carry on just fine without any life anywhere having the concept of "two"
The universe could carry on just fine without any life period. Indeed, it almost certainly did in our distant past. However, there is pretty conclusive evidence that there were quadratic relations among various physical quantities in the earlier universe, notably the inverse square laws for electrostatics and gravitation. It seems extreme to blame that on humans...
I will repeat one of my comment from yesterday [1] as we argued about the same topic.
I want to argue that whenever you think that there is some math already contained in the universe, it is actually you accidentally introducing it without noticing it. Look, trees! One, two, three, ... See? There are numbers in the universe. And look, I can add the number of trees in this wood and that wood.
It is of course tempting to conclude that the math was already there. But I argue that it was not, that it is you inventing it. It is your concept of object identity that defines what a tree is and when two trees are the same or different trees, you divide the world into different objects. It is your concept of grouping objects into sets. It is your concept of pairing the elements of two different sets to compare their cardinalities. It is your concept of including one more element in a set that defines what counting means. It is your concept of combining two sets that defines what addition is.
That are all things you can do with objects in the universe, but nothing of that is inherent. If you look carefully enough, all the things that look like math already exists in the universe actually require you making use of a lot of abstract man-made concepts. Many of them, like object identity and counting, are just so deeply ingrained in us that you don't notice that you are using them if you don't pay careful attention.
That is why I asked for the empty universe, I wanted to take away all the objects on which you could accidentally impose structure. If math were truly fundamental, if Platonism were true, then seven should still exist in an empty universe. I don't see how this would be true in any meaningful way.
If we added a mathematician to our empty universe, he could invent math. He could come up with the idea of objects and identity and sets of objects and he could think about operations with those objects and consequences of the definitions he made. But it would all have no relation to the otherwise empty universe, it would be a game of formal rules, mostly manipulating strings of symbols. If the mathematician had great imagination, maybe he could imagine actual things existing in the universe, how he could label them with those numbers he invented, and how he could visualize addition by moving things around.
In another comment I argued the same for your example of the inverse-square law. It is you who decided how to measure distance. It is you who decided to look at spherical shells around a charge. Nothing of this is inherent. I could easily make it an inverse-cube law by adopting a different measure of distance. I could decide to look at boxes instead of spheres and ruin the symmetry. It is certainly correct that there are more and less natural choices, but I don't think that is a good enough argument to conclude that some preferred math is intrinsic.
Particles flowing outwards from an isotropic source with constant velocity in straight lines in three-dimensional Euclidean space will generate a density field following an inverse square-law. But that is not intrinsic, it is just a useful description of the situations, by itself there are only particles traveling at constant velocity in straight lines. Systems with simple rules, especially all systems with chaotic behavior, can produce behavior with staggering complexity requiring some serious math to describe, but, by itself, there are just many things following simple rules and interacting with each other according to simple rules.
Give me a large enough pile of sand and I will use it to implement set theory on top of it. And then on top of set theory a huge part of math. Does that mean that grains of sand include most of mathematics? That does not sound reasonable to me. I believe people are confusing the ability of describing things with math or implementing math on top of something with the thing already containing the math. Also note that there often many ways to treat the same thing, are they all equally intrinsic?
That all seems very similar to the general question of existence. Do all books and pictures already exist, including those that will be written or taken in the future? An empty SD card certainly has the ability to contain them, I just have to flip the correct bits. To me it seem to simply stretch the definition of existence way to much to declare that everything I can imagine or everything I could create already exists. So being able to do arbitrary math with grains of sand or being able to describe what is going on in the universe with math does similarly not imply that math already exists.
That became way longer and repetitive than intended. On the other hand I repeatedly failed yesterday to convey my argument, so I will just leave it as it is.
I replied to one of your comments in a topic that was not [1] and not even obviously related. Additionally there was no argument as you did not reply. So I'm thoroughly confused by your implied context.
I'm a wishy-washy formalist and know some things about Boltzmann Brains as a theoretical diagnostic on de Sitter universes with slow-rolling inflationary fields, so I'm OK with your last two italicized paragraphs ("This is why I asked..." and "If we added a..."), but completely disagree with your subsequent non-italicized text.
In particular, geometry is a feature of nature. The Earth is round to a close approximation, and that is an observer-independent fact, not a convention.
> Does that mean that grains of sand include most of mathematics?
Of course not, but grains of sand include atoms and subatomic particles and those have features that are the same in non-sand substances in the various parts of the the solar system we have directly explored. We can describe those features in whatever language we want; but names, like sets of coordinates, are not features of nature.
Indeed, in the past century we've gotten much better at distinguishing between features which are observer-dependent (even if not obviously so) and features which are not.
In your sand argument, the pile of sand has some observer-independent features and some which are not. The number of the grains of sand in the pile is an observer-independent quantity, even if nobody actually counts them.
Formal models may be incorrect, but physical (as in not unphysical) ones attempt to describe real observer-independent features of nature. I don't think there are many physicists who, arriving at a novel prediction of a formal mode, would do anything other than "trust (or mistrust) but verify"; even the craziest model-building theoreticians I can think of are hungry for contact with nature via controlled experiment.
I'm disagreeing with the overly poetic language of "touching immortality" etc. (and German idealism... but that's another topic)
It doesn't have to be observed as some material manifestation to be a discovery. Notions can also exist outside human intervention and shape the universe.
Which is also why humans can discover such things (like pi) in parallel, without one discoverer ever having been in contact with the other.
Or why even aliens can discover the same concepts.
But if you require a more direct understanding I would say there is a perfect circle in every rotation
It would take an infinite amount of power to calculate every digit in pi, given pi itself is irrational, even while describing how to calculate it is rational.
I answered where circles are in our universe and you then try to change the original intent of omalleyt's question to mean a symbolic representation of pi
> It would take an infinite amount of power to calculate every digit in pi
Calculating pi is so simple life without nervous systems does it
Now, expressing that value in our limited yet overly verbose base ten symbolic representation would indeed require infinite recursion
It seemed to me you were changing the question that was being asked, and a question asked implies an answer, or 'goal post'
But if you were just developing new inquiry stead building off omalleyt then I apologise and would hope to encourage like critical thinking in the future
.. Thinking on it 'goal posts'.. and 'soap box' elsewhere in this thread.. lacks a timeless quality, I really should have left it out ;P
Representation by digits is just one representation. We have plenty of finite representations: http://mathworld.wolfram.com/PiFormulas.html
Base 10 notation also requires a calculation to get the concrete number: You have to sum and multiply and power a lot
(Imagine whole numbers described in base 10, now how you get that number of something?)
Yeah, that's not true. Many things in the universe have perfectly spherical or spheroid shapes, cross sections, or propagation patterns, like black holes or isotropic EM emissions or (more abstractly) light cones.
Pi is baked into the laws of physics in many locations as a consequence of the geometric behavior of our spacetime. It would also be baked into a lot of other plausible spacetimes. It's hard to quantify exactly what causes it to come up so much, but it's probably the fact that our universe is locally Euclidean and therefore uses the L2-norm for distance, which gives rise to circles/spheres as the unit shape and the extrema of a number of optimization problems.
Edit: "Scaled to the size of a beach ball, say scientists, the sun's equatorial bulge would be less than the width of a human hair"[1]
And, I assume the defects in the event horizon of a black hole are even more negligible.
[1]https://www.theguardian.com/science/2012/aug/16/sun-perfect-...
I'm not sure quantum mechanics and infinite precision are reconcilable
Basically trying to say that Pi, to me, doesn't appear to be a man-made construct. Any naturally occurring circle that's near perfect seems to illustrate that. Any deviation from a Pi-derived shape has plausible reasons for the defect, like the one you point out.
If something is "out there" as a specific concept as you claim, then it requires a perspective to isolate it as "a thing" which means you are basically claiming there must have been a creator.
If there is no creator then there is no perspective other than that of observers like us.
Unless you of course claim that the classical physical universe is fundamental not the quantum physical one.
This statement doesn't follow. They're entirely unrelated claims.
> If there is no creator then there is no perspective other than that of observers like us... the quantum physical one.
Stop trying to integrate pseudo-quantum physics into hokey philosophical arguments. You don't know what you're talking about, and it detracts from any value the argument might otherwise have.
Humans are not unique as "observers" in quantum physics. Air molecules appear to work as well. We haven't solved the measurement problem, but we know enough to say that it almost certainly doesn't have any bearing on arguments about platonic realism.
We can build mathematical objects that look something like clouds according to the observable characteristics deemed intrinsic to our definition of "cloud". We can't simulate real cloud systems with arbitrary fidelity because the the mathematical foundation we've invented leads rapidly to intractable computation. "But supposing we had infinite computation" is magical thinking.
Still, the universe gives us clouds. It's operating on different foundational principles.
Now, moving from sky clouds to things like electron clouds, we have the added problem of what appears to be "true" non-computable randomness (not just chaos) underlying it all. Many worlds doesn't rid us of the randomness problem but merely shifts it to the problem of deciding which branch we inhabit before/after an event.
We can still hang onto the notion of mathematics as discovery by recognizing it as a process of self-discovery about how our brains perceive the world.
Air-molecules doesn't change anything with regards to the claim that things exist "out there" non what so ever.
If your only beef is that you don't think I know anything about QM then I will live with that.
Perspective matters in terms of "proving" something outside of observation, by the assumed fact observation can change perspective.
In other words, if all of this is a single entity, it's going to be hard to prove that because bits would exist outside perception due to the cost of it being a recursive operation. One may not know oneself without recursion.
If something exist "out there" as a separate concept then it can only do so because it exist as a discreet entity.
Yet QM shows us that the world isn't fundamentally discreet but rather non-local.
We can observe because there is a discreet universe to observe within, but that does not make the observable universe the foundation of reality.
And so if you want to claim that something exist out there without there being a designer or without our perspective of reality being what turn it into existence then you need to explain how it relates to the entire universe not just the classical one.
No, it doesn't show us that. A number of collapse theories (notably many worlds) have non-locality baked in, but we don't know if they're correct.
Please stop trying to use QM in philosophy arguments. If you do, you're almost certainly misunderstanding what it entails.
Entanglement seems to be fundamentally non-local - unless you posit hidden variables - and does not depend on the interpretation.
Bells theorem shows that there is actually "spooky action at a distance".
So I am not using some arbitrary collapse theory in QM as a philosophy argument I am using Bells theorem which happens to be one of the most important experiments in the field of QM as I am sure you know better than anyone.
I would also say that it is possible we are all missing the point.
QM isn't outside the frame of reference of the scientific method though. It's as proven mathematically and empirically as general relativity is.
So the only irrational thing here would be to claim that the quantum world isn't as real as the classical physical world.
Agree on the last part though. we might (are most likely) all be missing the point
So I get you haven't appreciated Mahler and Bruckner. I can understand your position if you haven't. /s
It's important to remember that for all the books written about physics in layman terms, especially quantum physics, there comes a time when words don't cut it and only math allow you to continue the conversation.
I think our math education do fail in that way. Math essentially is a logic system where you set up premises and derive and make sure everything is consistent. That can be applied to real life such as accounting and physics, or it can be applied to an artificial game -- number theory is such a case. Our education emphasize on the latter part, and it is quite irrelevant to real life, and it is hard. I am in mid-life now and on reflection, fractions, for example, had zero use for my life, and I am in the career of scientific computing. What we need is a concept level of understanding of mathematics -- realizing that the key idea of mathematics is basically consistency. Most importantly, the math required in the real life is not hard -- it is necessarily intuitive. What the education we receive is the technical gaming part of mathematics, e.g. how to actually find the roots of quadratic equation. I don't really remember the formula now, but I now have an intuitive understanding of the quadratic curves and the meaning of roots. And the same goes for quintic equations. It is not hard at all. I roughly sketch out the curve and I immediately have a rough idea of where the roots are. And at work, it is simply an exercise in programming. It is hard following the way in our education, though.
The purpose of education is to show them that knowledge exists, but not the knowledge itself. Knowledge itself is misleading. It is more important to recognize what you don't know than to remember what you know.
To clarify, I personally do not hate math. And I respect the discipline of mathematics. However, doing math (the way our education system assumes) is just like doing music or art, or playing star craft, it is not for everyone.
PS: on further reflection, not only useless, the concept of rational number has been quite damaging in our understanding of the real world.
Students study varied subject matter but only one subjects' grade 'matters' and can be chosen by the student
Personally, I think our maths education suffers from a lack of freedom.. I do think it can be for anyone but only because I believe anyone can bring to it whatever they'd like, whereas education tells you what areas need to be studied and how
is competition the foil of expression?
For example, What would motivate a student to put towards any effort in their non-main courses if it doesn't affect their grade?
Edit: clarity and accuracy
The student could just coast through a course.. they would have to sit through all lectures and would be assigned problems and reading but only one course would track their retention
I feel this is basically already what is happening in education with the prevalence of students' creative ways of undermining the current grading paradigm
But with the ancillary benefit that a student would be freed to actually do the work to foster an interest in at least one subject
Also, the student could be free to explore other subjects in ways outside of the means of tracking retention
I was pushing my math research since I was a child but instead was forced to abandon it and had calculus forcibly replaced as 'how I should think about mathematics' even though I fundamentally disagreed with its concepts
If I was free from the stress of doing poorly from the standpoint of grades I could have focused my interest on my own mathematical insight and simply allowed my education to expose me to calculus
wherein I would have retained an appreciation for mathematics and developed one for the practicality of calculus practices
>"When you take a course in Euclidean geometry is not the teacher putting a... learning program into you? ...You enter the course and cannot do problems; the teacher puts into you a program and at the end of the course you can solve such problems. ...Are you sure you are not merely "programmed" in life by what by chance events happens to you?" - Hamming
I'm curious as to how different mathematics would be for two otherwise identical civilizations but with distinct average lifespans. In alternate Earths where lifespan is 40 or 200 years, what's the state of mathematics in 2017?
It's also interesting to note that many of the haters direct their hate towards rote learning tasks like arithmetic calculations and memorizing multiplication tables, which makes me think that they don't really hate math but rather the way it is taught. I'd hate math too if I saw it as following procedures blindly! Perhaps with better computer math tools[1], future students won't be forced to learn "manual labour" math tasks, and instead focus on "white collar" tasks like modelling and abstraction.
[1] SymPy is the best computer algebra system ever: https://minireference.com/static/tutorials/sympy_tutorial.pd...
It took me until college to realize that I was just coasting through math courses and not really investing myself in the subject. Perhaps in an effort to avoid those long-ago memories? Everything started to click in college. Interestingly enough, it was my enjoyment of programming and study of philosophy that really helped to push me. That, and sitting in a calculus class and realizing that I had to either dive in or be royally screwed.
It's a shame that such stories are so common.
It must have been that bitter flavor of failure that first inured me against mathematics...it didn't help that the rest of my K-12 education never even attempted to demonstrate the enormous beauty of maths. In fact, until I discovered calculus on my own terms between high school and college, I understood mathematics to be nothing more than the practice of applying rote formulas to arbitrary equations. There was no rhyme or reason to the quadratic equation...it was just one of many "rules" pulled from the mathematical "rulebook," and math was simply the practice of recognizing when this arbitrary rule applied, and then applying it.
Therefore, for most of my life, math was not seen as a creative or exploratory discipline, and in fact the very opposite: One's ability in math was completely contingent on their memorization of rules and simply following them. It was purely robotic, the domain of tightly-wound, uninspired automatons. It was for squares, not free-spirited creative souls like myself!
Though I am disappointed to think of the heights and wonders I could have visited by now, had I been given a proper introduction at a younger age, I am no less excited by the wonders ahead of me, and the many years I have left to explore them. :)
Using Coq, Idris, Haskell, Purescript, Mercury and k give me that sense still. Anything that takes a long and hard time to think about.
The ability to produce an infinite series of numbers (by adding another one) does not constitute a proof of existence of anything infinite outside one's mind.
Good philosophers, like Leibniz and the Indian tradition, which includes early Buddhists, have postulated this, but popular memes are much stronger.
The ability to notice some inherent patterns of natural phenomena, such as the ratio of a circle's circumference to its diameter, does yield this ratio to be immortal, or any other inferred constant.
Math does not exist outside people's minds (contrary to the popular discourse from the Zen And Art Of Motorcycle Maintenance). Only a few of what we call conservative forces and different forms of what we call energy (or matter if you like) which are different aspects of the same That, to which human categories are inapplicable.
It's a survey, and the link text is a quote of one of the surveyors which I only found by clicking view comments then ctrl-f'ing 'immort' which took me far down the list to find this:
Shane Magrath, 48, from Australia, has loved math since grades 9-12
“One day the universe will be a cold black emptiness. However that which is mathematically true will remain vibrantly true even if the universe fades to darkness. When we do mathematics we touch immortality.”
You rely too much on proof by assertion to support your ideology
> does not constitute a proof of existence of anything infinite outside one's mind.
You argue a lack of rigor but refuse to uphold your dismissal to your own projected standards
You even claim there are texts to support your position but choose to leave out a link for others to review
I am comfortably an opposing opinion to your opinion but one of the things that confuses me about those that share it with you is the 'outside the mind' element
Can you elaborate on how mathematical phenomena only exist inside the mind and if so how that proves mathematics fails to exist immortally
Computation is a finitary process, but its behavior is informed by infinity. Even if infinity doesn't "exist" in the sense that we could ever observe and comprehend it, we can see its shadow everywhere.
> The ability to notice some inherent patterns of natural phenomena, such as the ratio of a circle's circumference to its diameter, does yield this ratio to be immortal, or any other inferred constant.
I have to disagree. The fact that there are patterns to notice is a deep mystery of the world and prefigures mathematical truth. Things are different in some ways but the same in other ways. Trivial, but fundamental. The fact that we can find mathematics everywhere relies on this fact.
What is a deep mystery for me, is that life, it seems, does not have the notion of counting at all, at least on the level of cell biology. It is uses "pure functional" molecular structures as "code and data" and message-passing as its "language", but this is quite another topic.
This, BTW, is a hint to justify the principle that what cannot exist without an observer does not exist in principle. There is, obviously the notion of a distance, but not necessarily of space. There is notion of a process as a result of applied forces, but not necessarily of duration and time. Time is a derivative. Derivatives does not exist, physical phenomena do.
Nothing does not depend on the real world, to begin with. Especially, products of the mind conditioned by senses and a language.