James Lyons-Weiler 2/3/2021
I like Sir Roger Penrose. A lot. He’s a clear thinker, a consummate scholar. But I disagree with him on the position that mathematics is discovered, and I disagree with his observation that mathematics is “out there” in the world, not something we impose on reality. He thinks there is a sort of “meta-existance” in which mathematics exists, beyond the real. But I think he’s wrong. Let’s take a short journey together into why.
Mathematicians celebrate when they have found matches between the structures they study and the real world. But while mathematics is found in this part of the Universe, it’s due to the presence of mathematicians, who are, by definition, subjective. Mathematics is not built into the fabric of reality. It is a lens via which we try to capture the nature of reality. It is a language by which we try to describe the structures and processes of what we experience. Like a mirror, which reflects but does not contain reality, we can learn a great deal about reality using this lens. The very representational nature of mathematics means that the sense of “discovery” that occurs in minds of mathematician are examples of serendipity: brilliant minds confusing deductive necessity of finding a match between structures in mathematics and how the physical world works. They are confusing the outcome they are witnessing with discovery. For me, the preference for the beauty of “elegant” equations and the annoying habit of simplifying equations can decay the richess of causal explanations.
These are big thoughts for a biologist. So let me explain.
Any mathematician can define any term. Let x = 2. Or Let X = the sun. Or Let E = energy, M = mass, and c = the speed of light. Let us all continue then to agree that when we place a 2 above, and to the right of a number or a term, it symbolically represents that we square that number. Thus, E=mc^2 has meaning to those who learn the rules; they know that E=mc^x = E=mc^2.
The symbolic communication that occurs, the language of math, is clearly constructed. That is not debated. Perhaps we could say y = 2. Or instead of 2, let’s say that the symbol Ꭾ will represent “two of something”. So be it. This arbitrary assignment of meaning to symbols is the epistemological equivalent of the arbitrary meaning of words: they mean what they mean because we have all agreed – subjectively – to allow them to mean what we say they mean.
My friend Garth Stein, for example, author of “The Art of Racing in the Rain” and “The Cloven: Book Two” (and co-host of our podcast, “Uncanny Valley”) would still be who he is if his parents decided to call him “Raoul”, or “Steve”.
To me, however, and to everyone who knows him, the instrinsic “Garthness” of Garth, however, would not be quite the same if his moniker were not “Garth”. It’s apparent, self-evident, if you will, once you meet him, and get to know him. Your mind tells you that he had to be “Garth”; there is no other name that “fits” Garth the person was well as “Garth” the name.
Such is the nature of the human mind as it seeks subjective consistency. It feels inevitable. In fact, Garth himself feels inevitable. But it’s an illusion. It’s a construction of the inputs of our senses seeking a permanence to Garth. Because to know Garth is to love him.
Sadly, though, Garth was not inevitable. And neither was mathematics. Garth serves as a good example of the tautological nature of mathematical “truth”. Of course at the level of subjective assignment of symbols to numbers and operations, the entire language of mathematics is subjective, and, as I said, that’s not the issue. The fact is that the entire use of the entire area of mathematical epistemology is subjective; it is only that feeling of satisfaction that our toy models “fit” the world so remarkably well that we feel there must be some intrinsic truth inherent to the process of knowing by mathematics. It, like the inevitably of Garth, is an illusion which, if shattered, will make many sad. And it should. Like pondering the impermanence of a seemingly inevitable good friend.
The argument could be made that I’m being too hard, too exacting. Some, right now, I’m sure are furious at my musings. That’s understandable, it’s the first stage of mourning. Others are making inferences of thought experiments. “Surely, we expect we could show intelligent life forms from other solar systems our math, and they could show us their math, and still, 1 + 1 = 2 in their symbolic representation of 1 of their things “plus” 1 of their things will still equal two of their things, right?” they are thinking.
Yes, but that species’ decision and ability to make that representation is as equally subjective as ours; neither is correct, and both can be wrong.
Let me show you an example.
The equation 1 + 1 = 2 assumes a static nature not implied by the equation 1 + 1 = 2. Rabbits come to mind – a male and female rabbit implies >2. Now, mathematicians will defend math by saying the problem was not sufficiently specified, that it was a setup. Therein lies the crux of the problem: as a representation of truth, how do we ever know when symbolic representation is, indeed, fully specified. The statement “Given the local information captured, and, as far as we know, “1+1=2” is NOT inherently the same as “1+1=2”. Some mathematicians would say “that first part is implied”, and act as though the mental exercise of stating “clearly, look, guys, ‘Given the local information captured, and, as far as we know, 1+1=2’ is not that same as 1+1=2” is all some set-up. This begs the question: if we assume mathematics is rational, and real, on the basis that all unknown prior assumptions are allowable to make it fit new information, then clearly, mathematics can asymptotically approach the truth, but it can never, ever BE the truth.
Note of course that in the equation 1+1=2, or, in the case of some male and female rabbits, 1+1>2, the model also does include the food they eat, the water they drink, the air they breath. Two rabbits in a vacuum will always be two dead, but not rapidly decaying, rabbits. And sometimes 1+1=1 when it comes to rabbits: I’ve seen female rabbits kill male rabbits with home they don’t care to share their space. Reality bites.
For the rest of the universe outside of killer female rabbits, it’s worse. MUCH worse. For mathematics to BE the truth, it would have to reproduce, energy wave by energy wave, subatomic particle by subatomic particle, atom by atom, molecule by molecule, element by element, ion by ion, compound by compound… planet by planet, star by star, galaxy by galaxy THE ACTUAL universe. Some say mathematics could do that given enough time and (basically) an infinite amount of energy… but that would go beyond the realms of reality into a hyperrealm, a metaphysical universe within which universes could be contained, and their new universe would exist right next ours, a perfect mirror reflection of reality, i.e., another reality, captured in a hypothetical meta-universe.
If they succeeded, however, this new reality would have two universes in it, and I suppose then would all agree that 1 + 1 = 2 adequately and completely describes the new reality. In the very act of writing “1 + 1 = 2”, however, the universe(s) have now changed, and the definition of what “1” was and what the “other 1” was and thus was “2” is has also changed. So now, the ubiquitous “Oh, you didn’t fully specify the problem, it’s a set-up” defense of mathematics as rational is invoked. That invocation changes the universes once more, and thus the infinite chase where reality is always one step (or a trillion x trillion steps) ahead of symbolic representation continues.
Now, if you’re a mathematician, or a philosopher, you might argue “Well, Dr. Lyons-Weiler, you seem to have raised the bar, or moved the goal post” and a fair concern would (in kind) that I’m using English language to represent the problem of whether mathematics is ‘real’, and thus my critique or analysis is also therefore subjective and arbitrary and is not in any way foundational. This point I concede wholeheartedly; re-run the past evolution of life on our planet, neither I, nor you, nor this laptop would be here. My analysis would never exist; but then neither would human mathematics. The problem itself would not have been stated, but then again, neither would mathematics have been developed, and yet reality would (I am certain) be humming along its 10.4 billion-year trek, unfolding sans Homo sapiens, and, sadly, sans my friend Garth.
There are others who have different viewpoints on these matters, people with different and better degrees that mine, but I’ve studied the assumptions of math for sometime, and there are rules that are in place specifically because if they were not, the irrationality of math would become apparent. To save math, for example, we’re not allowed to divide by zero. One would think that such a rule, which is universally known, followed, and accepted is due to some dire consequence such as the emergence of a black hole. (The quote “Black holes are where God has decided to divide by zero” is attributed to Einstein). But no, the reason why we are not allowed this neat trick is not because we cannot grasp the idea of something being theoretically divided into an infinite number of things, that’s easy.., i.e., (warning!)
2/0 = ∞
and then of course
2/∞ = 0
These we can imagine, and math allows imaginary numbers. The reason why we are not allowed to divide by zero is because if we do, then not only
2/0 = ∞
1/0 = ∞ and
3/0 = ∞
1/0 = 2/0 = 3/0… ad infinitum
and all numbers would necessarily be equal to each other – proving mathematics itself to be irrational, imaginary or both.
We can try to imagine infinity all day long, and no one complains. I’m doing it right now. Infinity is safe. It’s so far away, it is so immeasureably and, paradoxically, unimaginably larger than every other number you can possibly state.
Such is the limits of our minds.
But then making each imaginable number, each knowable number roughly equivalent to each other, or, for all practicle purposes, numerical equivalents does not seem to be a stretch given the definition of and our understanding of infinity.
I mean, we can say, without blinking that x = 2, then we can say, compared to infinity, 2=3. Or something similar to that.
We can say, for example 2≈3. But that’s not strong enough. Because compared to infinity, 2 is by all reasonable measures not different from 3.
Since symbolic representation in math and language is arbitray, we could say
which reads “compared to infinity, 2 is basically the same as 3 by all measures” and then we can all move on from the awkwardness… we can, for the sake of making math consistent and rational, also reserve a special symbol for use when we do divide by zero (i.e.,
Whew. Still here.
As you can see, this exercise demonstrates that is it possible to preserve mathematics as a rational exercise by admitting that it is created, not discovered; that the rules of mathematics are made up by our species, and as far as we have come… our models are not reality. More importantly, that reality is not generated by models.
Does that mean we cannot “discover” models that are really useful? No.
I’ve invented a transcendental algorithm that can find an infinite number of equally good solutions useful for predicting health outcomes, or for diagnosis given a large set of biomarkers. The solution set does not converge; yet different runs of the software yields divergent decision rules that are equally good yet vastly superior in accuracy to traditional types of models inherited from, say, regression techniques. The method is generalizable and can be also applied, for example, to the task of optimizing a logistic regression in a manner superior to goodness-of-fit or least squares axioms.
I agree with Stephan Wolfrom in his descriptions of 50,000 possible combinations of mathematical axioms. So “sets of mathematics” do exist and can be discovered, but they have traditionally been constructed. and in the Wolfram super-set, meta-constructed. Mathematiccs has been self-limiting; Stephan says that he classify all past mathematicians based on the set of axioms they had, and can tell what their proofs would have and could have been if they had one or the other different sets of axioms. Stephan’s work shows that as smart as the smartest human beings are, we are but infants in the ocean of epistemology.
Let’s consider an important real-world example.
In The Order of Time by Carlo Rovelli, a theoretical physicist, the author marvels at how time cancels out of the important equations in physics. This, he insists, means that time does not play into reality in the way that our empirical experience tells us it must. The author, and in the audio version, Benedict Cumberbatch help the reader also marvel at how time merely falls out of the equations used by theoretical physicists and they help us peer over the edge of oblivion and helplessness in our inability to grasp the importance of this “fact” in our daily experiences. I hope that sometime we can explore the significance of whether time past, time present and time future all really do have the same fundamental properties (i.e., units, dimensions) that allow us to cancel these terms out during the derivation of Einstein’s (and others’) models in theoretical physics. All empirical evidence tells us that at a specific point in time, time past, present and future are fundamentally different; it is only the assumption that they are they same that allows us to cancel the term t out everywhere to create beautiful and elegant but misleading equations.
But then, when our conclusions about the nature of time are derived using equations that are solved allowing that time cancels out, our conclusions derive from our assumptions. which violate empirical experience. Something is wrong with the model, and perhaps we can construct a better model of the truth, perhaps by, as I have, assigning time past, present and future different units (dimensions) and viewing the relationships among the four dimensions of space, time, energy and mass in a new light.
I encourage you to watch Sir Roger Penrose and others discuss the issue of whether mathematics is constructed or discovered in the fantastic short (26 minute) piece in “Closer to the Truth” (below). I enjoyed especially the bootcamp review of the Mentalististic, Physicalistic, Platonistic (Realism), and Anti-Realistic (Fictionalism) views on whether mathematical objects exist to be found. The different viewpoints are well-represented. Even meta-mathematics is featured; the importance of that field of inquiry cannot be possibly understated. Stephan Wolfram is the Carolus von Linnaues of mathematics; his team is on journey to find, characterize and systematically classify all species of mathematics. For some odd reason, that makes me excited and gratified: we’ll have (as far as we know) a complete map of the knowable via mathematics. It will be an end and a beginning.
It is abundantly clear that for lack of a plausible mechanism, what we call math does not generate the universe. It describes it via axioms and proofs. And we have discovered math that fits reality well. The trashbins of mathematicians throughout history are filled with anti-math and missed hits; the solutions that did not fit mixed in with the rarer tossed good solutions. Some good models have fallen or were thrown into the trashbin and have not been recovered. Some have. From a signal processing perspective, mathematics is a detector of models that evaluate the reality of those models on how well they fit empirical experience. but they are models. So, it’s learning. It’s bootstrapping. It’s approximation. And as guys like Stephan Wolfram and his team map out, study and characterize all known possible mathematics, they will publish the structure of the universe of mathematical models. One thing is for certain: as exciting as that it, it won’t be a universe.
I vehemently disagree. I think you’re conflating ideas here.
Mathematics is a concrete reality that we discover. Whether the *nomenclature* for it is created, and whether there are created elements we *add* to the mix doesn’t change that.
Skunks exist. That’s an objective reality. Whether we choose to *call* them skunks, or make up some other name for them doesn’t change the fact that they *are*.
Likewise, core math concepts are inescapable. If you have one apple and you obtain another apple, you now have two apples. If we found another way to describe that or made up different words for it, it wouldn’t change the fundamental reality. The “oneness” of a single apple combined with the “oneness” of another single apple brings about the “twoness” of the apples combined.
Even the inability to divide by zero relies on reality that exists, independent of us, in the nature of what the manmade *symbol* zero represents — nothing. It’s an inherent part of reality that we can’t divide by nothing. There’s a sense in which you can add nothing to something, take away nothing from something. You can “have” something zero times. But you can’t divide something by nothing; that’s a meaningless concept.
The symbols and equations and language just represent what’s already there.
Just as a skunk exists, with or without a name or description.
Thank you for your comment Rachel. Allow me to rebut.
In my essay, I distinguish (perhaps obliquely) between the semantic construction of terms, operations, and the more fundamental problem. I brought those issues up to dispense with them.
I’ll agree that someone had to discover relationships that exist among interior angles of a triangle. They are just there. However, those who say that mathematics determines that reality have it exactly backwards. A practical infinity of triangle that ever existed by accidents of geometry of nature before humans evolved – in ice crystals, for example, had a set of relationships among interior angles. But that relationship – the reality so to speak – was never expressed or descibed in terms of “adding up to 180 degrees” until we created the construct operation of addition and the construct dimension of “degrees”. Nature works just fine without mathematics – many times. Saying that mathematics is “out there” waiting to be discovered conflates the model with reality. Composers construct symphonies. There are not symphonies “out there” waiting to be found. The set of all possible symphonies, of which there is an infinite, is a fantasy – an abstract representation of something that, in fact, does not exist. It’s liberating to view mathematics the same way, because we can transcend above axioms and see them not as ultimate truths but as useful assumptions – perhaps permanent, perhaps mercurial, pending new developments in ways of knowing that are not constrained by a set of axioms. “Everything we think is true is true until proven otherwise” is paradoxical if by “truth” we mean real. The solution is to stop thinking that models are real, or that reality is based on underlying models. A famous example of an axiom being overturned was Gottlob Frege’s Begriffsschrift. He had established a set of basic logical principles from which, he thought, all of mathematics could be deduced. He called his axioms “Basic Laws”, and one of which allowed the construction not just one set, but any set, provided you specified a rule for membership. This particular axiom was disproved by Bertrand Russel, who used Frege’s axioms to describe a counterexample known as Russell’s paradox. For a while, Frege’s model seemed to match reality. Because it did not, into the trashbin it went. It became what I call anti-math. It is through this repeated process of trial and error that “beautiful” and “elegant” solutions that seem to be so marvelous well fit to reality emerge. I can write an equation that describes the changes in gene frequencies in an evolving population. As much as I’d like to think it is reality, I cannot make my mind allow that, because I’ve traveled over ecosystems filled with evolving populations, and they are not p’s and q’s and r’s. The decades of work in genetics that modeled inheritance with modification, selection, genetic drift, and even gene x gene interactions and gene x environment interactions were grossly oversimiplifed (as beautiful as they were) because they did not include epigenetics. They could not include the complex biological pathways through which epistasis is mediated accurately because they were unknown. Sixty years worth of some of the most beautiful and compellingly elegant mathematics are buried in the Journal of Population Genetics, mostly all of which will never be explicitly tested, due to the the resolution with which we comprehend the increasing granularity – the larger number of parameters – and yet at this high resolution no one can derive even the beginnings of a singular mathematical framework that describes a human infant to the point where we can predict what that infant will do in the next few seconds – let alone the next sixty years. We can conduct simulations, say, for a single cell, but they diverge from reality on their own course rather quickly. No, our hubris tells us that our mathematics is the at the foundation of reality. Re: division by zero, if I can imagine something divided infinitly, I can imagine it divided by nothing as the inverse of that.
PS The skunk analogy, while popular fails because it merely shift the focus from something not real to something real. But I’ll offer that there is no mathematical equation, nor set of mathematical equations waiting to be discovered that is the equivalent of a skunk. If one is constructed, we will know it by its goodness of fit to a real skunk, and it will only emerge after many, many failed anti-math models of skunks are thrown away for failing to adequately represent a skunk. If someone derives from existing axioms an equation that describes a skunk with ever throwing away failed attempts, without constructing new representations or discovering new relationships among parts of skunk ee do not yet know, I’ll be stunned. But, even then, the equation will never be a true skunk.
Yes i totally dis agree also, I would have to write a huge essay to counter some of your points. but just to allude to 2 or is that ( 1+1 ) yes it is. Here or anywhere else, even if no observers existed. you mention you found a better solution to the “least squares solution” Gauss proved that if a variable is normally distributed the least squared solution is the best possible solution, involving the least error. But not all data is normally distributed… but that doesnt stop people applying it as an approximation, because they do not know what the underlying distribution is. also they may have a paucity of data so it could be possible to find a better fit… ( but can you prove theoretically its a better fit, or it just looks like it to you.) Also you dont know the underlying distribution. So its impossible for you to theoretically prove you have a best fit solution. As for infinity, there are various types of infinity know, I suggest you red up on Cantor. Also you say mathematics is useful (only ) if it can model or represent physical reality, however the vast vast majority of mathematics has no know application. Its true in its own right, independent of sometime much later found physical application. eg complex numbers later found to be useful in quantum physics.
Thanks for the comment, Luke. Re-stating 1+1=2 as a universal given truth makes it axiomatic.
The superiority of the model I mention is context-dependent (heterogeneous information distributed among variables)
and I admit it is not demonstrated to be universally superior. However, in the canonical setting in which is assumed that every model variable has an optimal parameter such that each parameter contributes to all entities being predicted, my algorithm will be superior from first principle, and empirically.
Thank you, Dr. Lyons-Weiler. Much, much food for thought here. I think I’m pretty well persuaded that mathematics is a descriptive construction developed in an attempt to make sense of reality. But it is truly intellectually beautiful. My greatest growth as a teacher came from teaching math (I taught all the subjects), a universal language. The interviews were wonderful.