I am angry with my friend; he has betrayed a secret of mine perhaps, or maybe instead he has formed an intimacy with persons he knows to be my enemies. I start to view his behavior as treacherous, his character as disloyal. As I am convinced that my anger is justified, given the circumstances, it grows all the more intense. I begin taking my revenge is small ways, disparaging him in conversation with mutual associates, or waiting a little longer than usual to return his phone calls. As time goes by, however, I reflect on my own behavior; I realize that it is my own character which has not come off well in the event, that I am behaving in a petty and malicious manner. I grow ashamed. I remember something about the importance of forgiveness, and I recall also the many kind favors my friend has performed for me in the past, and my obligations to him on that account. Eventually, my indignation wanes, and I reconcile with him as though no breach in our friendship had ever arisen.
I have tried to imagine a scenario of a rather mundane variety, such as must transpire many times in the lives of most people. What I would ask the reader to consider is how deeply the experience I have described is conditioned by language. It would be a mistake to suppose that, in this case, language traces or recreates the experience, either to oneself in introspection, or to others in narration, as though the emotions and the memories and the intentions were one thing, and their verbal description something else entirely. Rather, the linguistic understanding of the experience permeates and shapes that experience down to its roots. I regard my friend’s behavior as disloyal because I have some concept of “loyalty” in mind; such a concept requires language for its formation. My conviction in the “justified” status of my anger, too, presupposes some linguistically delineated boundary between right and wrong; it is the perception that my friend’s behavior falls on the wrong side of this boundary that fuels my wrath. Later, my shame arises from the thought that my own behavior would fit adequately under the categories of shameful actions; again, those categories can only be grasped through language. My memory of my friend’s favors as obligating actions likewise stems from some linguistically-propounded notion of an obligation. Throughout my experience, language determines thought, which in turn arouses emotions, conditions memories, and alters purposes and actions. Language determines the experience itself; attempt to excise language from the experience, and you will find that there is no experience left, at least none corresponding to the one described. There is no indignation, no recollection of a debt of gratitude, no sense of shame, where there is no language to determine these things.
To say that our experience is determined by language is to say nothing less than that our experience is determined by a particular system of symbolic representation. Our experience is constantly mediated through the symbolism of language, and this affects the tenor of that experience at its most fundamental levels. The attempt to isolate what in our conscious encounter with reality has been determined, even ever so slightly, by linguistic interpretation from what has remained wholly free from such determination would be an extraordinarily difficult task, and, in the end, we would likely discover that very little in our experience has escaped such determination. In consequence, any account of human life, and common human activities like political governance or art-making, must be an account of linguistically determined phenomena. Any methodology applied to human life must be capable of encompassing that full range of linguistic determination, without remainder, or it is not adequate to the task of rendering human self-understanding.
We live in an era when science is purporting, with ever greater brashness, to offer us the definitive accounts of human life, and general human practices like worship or law-following. The assumption that the methods of science are adequate to this task is extraordinarily widespread. Yet equally widespread is the ignorance of what exactly those methods entail. When modern science developed in the seventeenth century, the new and distinctive thing about it, as everybody understood at the time, was its capacity to render mathematically demonstrable laws of nature. This is what gave its partisans the assurance that the explanations of science were more conclusive – more true – than anything its rival and predecessor disciplines had ever delivered. Language would become increasingly suspected as a menace to clear thought (as in Bacon’s famous Four Idols), and the linguistically-grounded humanities dismissed as so much interminable and inefficacious wrangling. Galileo, for instance, in his Dialogue on the Two World Systems, will claim explicitly that the humanities do not state truth; nature, as he famously asserted, is a book written out in mathematics, and only the new mathematically based science was up to the task of interpretation.
Perhaps the best study of this revolution in thought is to be found in E.A. Burtt’s The Metaphysical Foundations of Modern Science. As early as Kepler, Burtt notes, the assumption that “perfect knowledge is always mathematical” was in place, an assumption that would become the root thesis of Descartes’ speculations, “that exact knowledge in any science is always mathematical knowledge.” Newton based his entire physics on “the certainty of mathematical demonstrations,” and the success of his system seemed to vindicate to one and all his opinion that “science is the exact mathematical formulation of the processes of the natural world.” Thus modern science marched into the world heralding a new standard of truth in the mathematically demonstrable; from then on, only what could satisfy this standard was to be accounted true.
If we examine those new scientific theories of humanity I alluded to, we will find that they more often than not rely upon the heavily mathematical framework of neo-Darwinism and evolutionary psychology. Statistical genetics, game-theory scenarios and their accompanying calculations, probabilistic analyses of behavior such as derive from the theory of inclusive fitness – these things comprise the indispensable tools of modern biology. I do not wish to dwell here on the respective problems attending the employment of this methodological machinery. I only wish to point out that in employing this machinery, contemporary biology is conforming perfectly to the original impetus of modern science, proving that the pursuit of “the certainty of mathematical demonstrations” is still its dominant enterprise.
Whatever else science is then, and whatever else its methods entail, they do most certainly entail the exchange of one system of symbolic representation for another – an exchange of the linguistic system of symbolism for the mathematical. So the question that must be answered, prior to the formulation of any specific scientific explanation of this or that form of human experience, is whether or not the system of mathematical symbolism – a system comprised of numbers rather than words – is sufficient for the representation of the entirety of human experience. We have already seen one condition that must be satisfied in order to answer this question in the affirmative; since human experience is inextricably linguistic, down to its roots, we must be able to translate any potential linguistic statement into a mathematical statement. To put the point another way, mathematical statements must be able to represent everything that linguistic statements are capable of representing. I reiterate the point I made at the outset; human experience is not merely represented by language, but profoundly determined by it. So if there is something in language which cannot be fully represented by mathematics, then there is something in human experience that cannot be represented by mathematics; that means that there is something about human experience that science cannot explain. And if that something constitutes a considerable facet of human experience, then science is severely limited in its capacity to explain human life.
Is mathematics capable of representing whatever language is capable of representing? Are all, or even most, linguistic statements translatable into mathematical statements? Quite clearly, no. This is a point so obvious that I suspect most readers will grasp it intuitively. Return to the scenario I sketched in my intro; what would be the mathematical translation of “my anger is justified” or “my behavior has been shameful?” Again, one cannot remove such statements from the experience, because they have shaped the experience fundamentally – they are an irremovable part of the experience. But if mathematics can find no way to represent this part of experience, then science too, is impotent to explain the experience, or pronounce causes of the behavior bound up with the experience. Is this a major, even fatal, limitation of science? Consider how much of the experience, as described, is constituted by the various permutations of desire – resentment, and shame, and forgiveness are all forms of desire. The more we reflect on the matter, the more we will recognize how basic is desire to our subjective encounter with the world, and how much of our experience entails some impulse or other of desire. Yet numbers, in their austere objectivity, have no capacity to represent desire in any form. This is most certainly a debilitating defect in the methodology of science, as it is applied to human life, for human life is little more than a frantic melee of desires.
How is it, then, that such an obvious defect in scientific methodology continues to be ignored by hundreds of scholars, who continue to churn out thousands of pages a year, all purporting to offer the public scientific accounts of this or that arena of human life? Mostly, by ignoring the imperative which emanates from science’s own standard of truth, of finding “certainty in mathematical demonstration.” Precious little of what is published under the heading of science dealing with human behavior makes any pretense of formulating mathematical laws capable of covering the forms of behavior examined. This of course does not preclude their authors from assuming the scientific imprimatur of “certitude” and “conclusiveness,” which had always been understood as stemming precisely from the capacity to produce such laws; such authors are like bankrupts passing bad checks, drawing on a fund of objective truth they have not bothered to capitalize. Alternatively, as I noted, many neo-Darwinists and their disciples have taken to adopting mathematical descriptions of human life, brazenly following this approach to its most ridiculous conclusions, as in Richard Dawkins’ contention that “we simply expect that second cousins should be one-sixteenth as likely to receive altruism as offspring or siblings.” Then there is the increasingly large number of theorists who essentially deny that there is any such thing as human experience to be explained, who espouse a materialist dogma which effectively rules out the reality of consciousness or subjective experience. Of this approach there is little more to say other than that when a man believes such things, he is ready for the nuthouse – or the university. All together, these various strategies amount to a vast epidemic of denial about what scientific methods are capable of accomplishing, and what they are not.
To be sure, this denial is nothing new. No sooner had modern science leapt into the world, boasting its early and indisputable successes, than thinkers like Hobbes and Gassendi attempted to turn its methods towards the creation of a new theory of human nature and human life. They ignored the fact that science had achieved its success precisely because, in adopting the system of mathematical symbolism for its researches, it proceeded according to a willful austerity, purchasing its accuracy and conclusiveness at the price of compendiousness. Three centuries later, and with an almost unfathomable advance in our empirical knowledge, the theoretical impediments to a scientific understanding of ourselves remain the exactly what they were: mathematics cannot translate language, and our experience is incorrigibly linguistic. The prerequisite for rescuing Western thought from its present doldrums is a frank acknowledgment of this point, and a framing anew of our search for self-understanding according to the methods of the linguistic disciplines – that is, the humanities. At some point in time, mankind must awake to this evident imperative. Why not let that moment be now?
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Mark A. Signorelli’s personal website can be found here: markanthonysignorelli.com
“…the question that must be answered, prior to the formulation of any specific scientific explanation of this or that form of human experience, is whether or not the system of mathematical symbolism – a system comprised of numbers rather than words – is sufficient for the representation of the entirety of human experience…”
Sufficient for representation *to whom*? If the whole world is math – or, at least, truthfully describable by math – then the answer to your question is yes *even if* we ourselves would find such a representation to be confusing or distressing. So far as I can see your only argument against the mathematicization of reality is that it feels weird, but this is about the least convincing argument in the world of philosophy.
I am not quite sure what your point is. I suppose it makes a huge difference to one’s ability to think about things how fluent one is, whether one makes many associations naturally or instead has to struggle to remember what each symbol means as it arises. And no doubt almost everyone is vastly more fluent in their native language than in any mathematical symbolism. So if you mean to say that lack of fluency hinders representation of some concepts in mathematical symbols, I suppose it must be granted. But it didn’t seem like that was the point you were making; if insufficient fluency is the problem with using mathematics to talk scientifically about human experiences, it would seem the solution is greater fluency, not abandoning mathematics in favor of tools which have produced virtually no meaningful progress over the course of millenia. Indeed, the lack of progress using those tools suggests that perhaps we don’t know what we’re talking about when we use our native languages to describe our experiences; as Hegel said, what’s familiar is absolutely not understood by being familiar.
And certainly any strong version of your claim is wildly false, as Eli points out on his blog. Any claim in any natural language can be expressed in mathematical symbolism using a suitable code (I modestly suggest unicode as a possibility for English and many other languages).
Since we are on the topic of science, let me propose the following law of nature – when reading through comment boxes, the brashness of assertion on the part of the comment writers will be in inverse proportion to their understanding of the topic at hand. Need empirical evidence? Ladies and gentleman, Eli, the Rusty Philosopher.
Mr. Rusty thinks he can refute my argument by pointing out that numbers can stand for letters, like “1-A; 2-B, etc.” and therefore mathematics can be made to represent whatever language represents. In all of the objections which I foresaw to my essay, I certainly didn’t anticipate anything this stupid. Of course, numbers can be made to represent letters; anything can be made to represent letters. We could say “two diamonds – A; three clovers – B, and so on,” and write out our thoughts accordingly. But in this case, we are just representing one system of representation with another; we are not representing reality. And what I clearly meant (do I really need to say this?) is that mathematics cannot represent nearly as much of reality as language. Notice, I didn’t spend much time belaboring this point; I simply noted that it was “obvious.” And it is – to anyone not thoroughly disingenuous. Sure, we can use the Rusty Philosopher’s table and say that “694812” represents anger, because it represents the letters “a-n-g-e-r.” But take away the middle level of representation – that is, take away language, and make the numbers represent reality itself – and how does “694812” represent anger? It doesn’t, and again, this is perfectly obvious. And I haven’t even begun to make the point that a linguistic proposition is not just, or even primarily, words, but syntactic organization as well, and that mathematical equations cannot model this feature of language. There is only so much sophistry one can dismantle in a combox.
But by all means, let the Rusty Philosopher sneer away, and boast about how he throttled dumb old Signorelli with this garbage. That way he can win the plaudits of his materialist friends, people like Mr. Boyden, who thinks that humanistic learning has brought no “progress” to mankind throughout the ages (well, yes, if you don’t consider Western civilization to be progress). It is only too plain to see that Mr. Rusty’s entire identity is wrapped up in the illusion that he is one of the enlightened ones, who, through the aid of science and modern reason, has soared to rarefied heights of wisdom the rest of us can only wonder at. To be forced to acknowledge that everything he believes is a load of bupkis would absolutely shatter his self-esteem, and I don’t want that kind of psychological cruelty on my conscience.
I wouldn’t presume to elevate it to a law of nature, but it is striking how often “it is obvious” as used by philosophers means “I’m hoping you won’t notice I don’t have an argument here.” But perhaps I’m being uncharitable. Still, since you think it is so obvious, it should be easy for you to explain it to me. First, let’s clear away a misunderstanding. The reference of the word “anger” is the emotion anger. The reference of 694812 in the code is also the emotion anger. It isn’t the word “anger”; the look-up table doesn’t tell you that the references of the numbers are letters, it tells you how to find the reference of the numbers if you have greater facility with English than with the code. That I need such help says nothing about the reference of the number in the code; I also need help, say google, to tell me θυμός refers to anger (assuming google hasn’t misled me; I don’t know Greek). That obviously doesn’t mean θυμός doesn’t really refer to anger. I’m sure you weren’t really confused on this point, but meant to make some other point about representation; I just thought I’d be completely explicit to clarify where I think you need to explain more.
You also mention syntactic organization, which makes it increasingly difficult for me to be charitable. You think there are aspects of syntax that can’t be represented mathematically? Really? That could really only be the case of Church’s thesis were false, and if you can give a good reason to think that, publish it and become a legend!
I expect that’s not really what you think is the crucial point either, though. So please, explain what you see as the additional elements of representation beyond successful reference and syntax, so that the rest of us may see your obvious point that English contains those elements and mathematical symbolisms do not.
The point is obvious Mr. Boyden, and frankly, I think you and your buddy are being disingenuous in pretending not to notice that language has a far wider frame of reference than mathematics – that the meaning of statements like “I enjoyed the play last night,” or “King Charles was a prudent man” cannot be expressed mathematically. If you believe otherwise, show me the equations which express the meaning of these sentences (and don’t simply represent the sentences themselves). You are also being disingenuous when you say that, according to the Rusty Philosopher’s table, the number 694812 refers to the emotion of anger, and not the word “anger.” What table are you looking at? The one I’m looking at has numbers next to letters; ergo, the numbers stand for letters. In what world does 694812 refer to an emotion? Your example of the Greek word does not work; it does mean anger for a set of people, the set of people who speak Greek. But 694812 does not mean anger for anyone in the world. Now, I suppose you and the Rusty Philosopher could get together and decide that this number will from thenceforward represent anger, that 694823 will represent resentment, etc. But what you will be doing in that case is trying to form a language, and one that will prove wonderfully clumsy.
I know nothing about “Church’s thesis;” you can enjoy your superiority over me in that respect. But I do know that, so far as the creation of meaning goes, syntax functions very differently in mathematics than it does in language. The equation “3+4=7” means exactly the same thing as “4+3=7.” But, as any good reader of poetry can tell you, “Something wicked this way comes” does not mean the same thing at all as “Something wicked comes this way.” And until you can make that difference go away, I set you and Church (whoever he is) at defiance on this point.
You wish to know the crucial point? I think it was clearly stated in the original, but I’ll try to state it again. Science relies on a unique standard of truth, grounded in “the certainty of mathematical demonstrations.” This must mean that everything which science presumes to explain definitively must be capable of representation in mathematics. But mathematics is, by its nature, a deliberately truncated symbolic system; it does not represent everything in human experience, and is not intended to. Science began in the 17th century by consciously limiting the scope of its inquiry, in order to achieve more conclusive results; that’s why it adopted mathematics as its primary symbolic system. Now, in our time, science is declaring its capacity to pronounce on the entirety of human experience. But it cannot explain the entirety of human experience until it can represent the entirety of human experience. And, as already stated, mathematics, its chosen symbolic system, cannot represent the entirety of human experience. So the grandiose claims of modern scientists about the scope of their theories are impossible to believe. Get it now?
Thank you for granting the point that 694812 can refer to anger rather than “anger;” I hereby stipulate that that is the intended interpretation for all my uses of the example (I already did this, of course, but you ignored me the first time).
On your second point, because order doesn’t matter in addition does not mean that mathematics is incapable of representing anything in such a way that order matters (consider 3<5). A common (though not mandatory) view takes sets to be fundamental to mathematics, and the method for making ordered sequences of arbitrary size and complexity from sets is well known to anyone even slightly familiar with set theory. So there's no need to ignore order in a mathematical formalization; if your poetic sentences differ in meaning depending on the order of their elements, represent them by suitable ordered n-tuples of mathematical elements. It is clear that you radically underestimate the expressive resources of mathematical symbolism, and it is becoming clear that this is because you have a very limited understanding of mathematics.
You also don't seem to understand the role of mathematics in science. Your original post is full of strawmen, but I grant that once or twice you may come close to actually criticizing someone deserving of criticism. But you never do so for the right reason. If someone fails to describe something using mathematics, this is wildly insufficient to reveal a limitation inherent to mathematics, since this happens with great frequency for one of two other reasons; because their understanding of mathematics is insufficient, or because their understanding of what they are trying to describe is insufficient. And, of course, such errors are also common when English is used to attempt to describe something, which is the point I made (though perhaps with a little excess rhetoric) in my first comment.
Mr. Boyden,
Are you serious? “Thank you for granting the point that 694812 can refer to anger rather than “anger?” I don’t grant this at all. I explicitly said that this was wrong, that the number refers to the word, not the thing, and that is the whole point of the Rusty Philosopher’s exercise. If you can misread such a straightforward assertion, no wonder you are having trouble grasping my point.
Do you really think the difference in the two sentences I indicated can be adequately explained by “suitable ordered n-tuples of mathematical elements?” Or are you just trying to see how far you can push this silliness? My simple point is that syntax operates differently in mathematics than it does in language, and nothing in set theory is going to call this into doubt. Please note – you wrote that the different “sentences” could be represented by the mathematics. But how can the different “meanings” be represented by the mathematics? You really don’t understand the difference, do you? As for my own limited familiarity with mathematics, I plead guilty. But I know enough to know that, as Thomas Reid wrote, “mathematical evidence is an evidence sui generis, not competent to any proposition which does not express a relation of things measurable by lines or numbers,” and that the portion of human experience which does not consist of “things measurable by lines or numbers” is far the greater part.
Your last paragraph makes it clear that you still don’t get it. The limitation of science I pointed out is a theoretical limitation; it has nothing to do with the practical defects in this or that scientist’s mathematical abilities. The role of mathematics in science is to be found, first and foremost, in the scientific standard of truth – ie, “the certainty of mathematical demonstration.” Nothing can be mathematically demonstrated that cannot be mathematically represented. And the full range of human experience cannot be mathematically represented.
“…what I clearly meant (do I really need to say this?) is that mathematics cannot represent nearly as much of reality as language. ”
Mark, either you don’t understand the meaning of the word “represent” or you’re lying. You yourself agree that math can represent ANY linguistic statement – why, then, can math not represent anything that language represents? Representation is transitive, after all.
“But take away the middle level of representation – that is, take away language, and make the numbers represent reality itself – and how does ‘694812’ represent anger?”
Is this supposed to be a serious objection? The phonemes (or letters) of “anger” are not somehow ontologically connected to the feeling of anger. They, like 694812, are just arbitrary symbols that we’ve chosen to use for convenience. Surely you know this, Mark. Why, then, are you pretending that math is inferior for working in the exact same way?
“…a linguistic proposition is not just, or even primarily, words, but syntactic organization as well, and that mathematical equations cannot model this feature of language.”
Oh, bullshit. How do you think your computer knows to put a squiggly green line underneath grammatically incorrect sentences? Because it models the syntax of English mathematically, is how.
Mark, I’d love to engage in a rational conversation with you, but evidently that’s not going to happen. You obviously have no idea what math is, let alone language. For example, you’ve said repeatedly now that “the full range of human experience cannot be mathematically represented” – but the same is true of language! We devise new words and phrases *constantly* because the ones we have aren’t (and will never be) enough. Why, then, does that not also disqualify language?
The only answer I can come up with is that you’re so married to this idea of “scientism” that you can’t be bothered to learn anything about math or science in the first place. As I said, I’d love to have a rational conversation about this, but if you’re more interested in fighting for your cause than you are in learning the truth, then there’s nothing to be done. I just hope you acquire some humility somewhere so that you stop embarrassing yourself like this.
Mr. Signorelli, by all means keep embarrassing yourself, if it leads to more interesting articles like this one.
As for Mr. Boyden and the Rusty Philosopher, all I’ve been able to discern in your comments is the argument that anything linguistic can be expressed numerically – which, of course, any seven-year-old with his first decoder spy ring already knows. (Full disclosure: I was always more partial to disappearing inks, myself.) And which, is not the same thing as saying anything linguistic can be expressed mathematically.
You’re a real piece of work, Rusty. The arrogance and pugnacity with which you come on here and spout your nonsense are astonishing enough, but when you turn around and admonish me to show humility, its just too rich. Its really quite simple, Rusty: produce the set of equations that mean what the “Ode to a Nightingale” means, or anything like it. Otherwise, pipe down.
You wrote: “The phonemes (or letters) of “anger” are not somehow ontologically connected to the feeling of anger. They, like 694812, are just arbitrary symbols that we’ve chosen to use for convenience.” I suppose you think you’re saying something important when you write this. You’re not. Of course language is conventional. All symbolic systems are conventions (unless you’re William Blake). Language, by convention, represents the whole range of human experience – memories and desires and anxieties and emotions. I wrote in my essay that our experience is intrinsically linguistic; conversely, our language is intrinsically experiential. We live in our language, and infuse it, by convention, with a range of significance that covers the whole of our experience. This is not to say this representation is perfect; again, you’re pompously throwing out a truism that has no bearing on my point. However imperfectly, language is a set of symbols that, by convention, has come to represent the full range of experience.
Mathematics, on the other hand, is a set of symbols that, by convention, has come to represent only a portion of our experience, namely, whatever has quantity or can be measured. Of course, you could decree that numbers are going to stand for certain words, like you did with your silly little exercise. But then you’re not doing mathematics anymore. Change the conventions, and you change the symbolic system. And this is proved by the fact that, as I noted, you could exchange numbers in your chart for any other group of symbols, to do the same thing. If you don’t believe me, start talking with your neighbors in the code you made up, and see how many people comprehend you.
You are the one who is being obtuse about representation, and the world of difference between a representation of another symbolic system, and the representation of reality. Take away language, and mathematics cannot represent emotion, but take away mathematics, and language can still represent emotion. That’s because it has a larger field of reference. As I wrote in my essay, “mathematics cannot translate language.” Do you understand what translation means? It doesn’t mean that one set of symbols can be made to stand for another set of symbols; it means that one set of symbols can stand for reality in a way similar to the way another set of symbols can stand for reality. When you translate Pushkin into English, it doesn’t mean you are devising a code which compels you to continuously go back to the Russian. It means you are taking one set of symbols and replacing it with another one signifying the same thing, or something very close, and the prerequisite of such a task is that both symbols have a sufficiently similar range of representation. That’s why Pushkin can be translated into English, or Japanese, or Swahili (albeit, very imperfectly). But not math.
As for your accusations that I’m a liar, an ideologue, someone incapable of carrying on a rational discussion – well, all I can do is throw those same accusations back in your snarling teeth. Actually, I have to hold myself back here – I’m pretty sure that FPR has a courtesy requirement for its comment section, which prevents me from calling you the sort of names you deserve to be called. So I’ll say this instead: I have repeatedly argued that humanistic learning is the proper road to self-understanding. I think self-understanding is important, and the lack of self-understanding disastrous in its consequences for character. When you come on here assuming an air of condescension to which your very slight abilities obviously do not entitle you, when you show yourself to be one of those who, as Socrates would say, “thinks he’s something when he’s really nothing at all,” you illustrate, in superlative fashion, the ugly, corrosive effects which a total lack of self-understanding has on one’s personality. All the best.
“Mr. Signorelli, by all means keep embarrassing yourself, if it leads to more interesting articles like this one.:”
Thank you, Tony A., for that thoughtful contribution to this debate. Now please return to your usual haunts at Huffpo or the Daily Beast, where comments like that actually pass for argument.
Mr. Signorelli,
Wow. Sorry about the poor attempt at giving you a compliment. Of course, I was being sarcastic, inasmuch as I don’t at all think you were embarrassing yourself (as in, “well, if this is what others call embarrassing, I’m all for it, please keep writing”).
As far as arguments go, I’ll readily grant your latest comment was much more polished (and longer) than mine, but I was trying to point out the basic issue that when you say “mathematics”, you are talking about an entire system of symbolic representation, which includes more than just the set of symbols used (e.g., numerals, etc), while Eli’s and Mr. Boyden’s arguments (as far as I can tell) only get as far as swapping out one set of symbols for another, and therefore don’t address your argument.
In other words, I was agreeing with you. Sorry for being such an idiot.
“Fire” was the only way the brilliant mathematician, Pascal, could translate his mystical God experience. Perhaps even Pascal was limited in decoding certain experiences, in language or math. So for someone far less capable, like myself, I boil the thing down by replaying a similar historic query. I recall C.P. Snow’s 1959 essay concerning the communication rift between the schools of humanities and sciences. Like Mark Signorelli’s FPR piece, it caused a bit of a ruckus (understatement).
Tony A. –
I see now. It was my mistake, so I’m the one who owes you an apology. Please accept it.
And thank you for the compliment.
My two cents, math gives us solutions, or bleary-eyed despair or, worst case, economists; language lends meaning, or the second thing – or worst case – deconstructionists.
Not certain what side I’m pulling for.
“produce the set of equations that mean what the ‘Ode to a Nightingale’ means, or anything like it. Otherwise, pipe down.”
Ever see the Monty Python sketch where they do ‘Wuthering Heights’ in semaphore code? Of course it could be done. But what’s the point? Would it still in any true sense be ‘Wuthering Heights’? Really, is this so difficult to figure out?
Rob G., I still wouldn’t say that that sketch had anything to do with mathematics, but I agree with your more general statement of “what’s the point?” The absurdity is part of what makes the sketch work. Another similar example is in Douglas Adams’ The Hitchhiker’s Guide to the Galaxy, where the Deep Thought supercomputer gives the ultimate answer to the ultimate question of life, the universe, and everything as “42.” Again, the joke is funny because of the absurdity of trying to solve the meaning of life as if it were a mathematical equation, as well as because of the utter seriousness with which the attempt is treated by the characters in the book.
But, as the essay shows, the fact that it’s absurd doesn’t stop people from trying. One prosaic example – I once worked for a sales manager who told me his only measure of whether he had a good year was whether he made more money than the year before. Of course, money is a useful and necessary means of measurement in the business world, but he had (or at least professed to have) no other vocabulary with which to evaluate his career. Was he honest with his customers? Fair to his sales staff? Selling a worthwhile product? Did the fact that he made more money than the year before justify his being on the road and away from his wife and daughters for weeks at a time? Now, I think that he was in fact honest and fair, sold a worthwhile product, and tried very hard to be a good father and a good husband. But, none of those virtues were derived from the mathematics of comparing last year’s earnings with this year’s.
A question for Mr. Signorelli (or anyone else who cares to address it) – you say that “our experience is incorrigibly linguistic.” Why is that? For the record, I agree, but if pressed to say why, I doubt I could come up with anything other than the Word was made flesh, not an equation or algorithm. I also realize that it’s probably way too big of a question to address in a comment or essay, and would be equally satisfied with suggested reading material.
(Also, no apologies necessary – friendly fire is always a risk when one strays onto the battlefield. Huffpo? That one hurt. The Daily Beast? I haven’t the foggiest what one is, but I guess it’s a good thing I don’t have one. Well, come to think of it, maybe I did, that year I gave up coffee for Lent. . . )
“Do you understand what translation means? It doesn’t mean that one set of symbols can be made to stand for another set of symbols; it means that one set of symbols can stand for reality in a way similar to the way another set of symbols can stand for reality.”
So this can have one of two interpretations. Either you mean that math can’t “stand for reality” because math can’t be used to symbolize things or you mean that math can’t “stand for reality” FOR US because we don’t speak in math-language. The first of these is false and the second is nothing more than “it feels weird.” This is, again, about the least convincing argument in the history of philosophy (and is, at any rate, irrelevant to your overblown claims about what science can and can’t do).
I really don’t know why you won’t just educate yourself on this topic, Mark, but your bluster and your silly nicknames don’t impress me any and won’t make you any less wrong.
Eli,
I don’t see anywhere where Mark has said that math can’t stand for reality – only that math is by its nature (or by convention, if you prefer) limited to “whatever has quantity or can be measured.” If, then, there are things that do not have quantity – such as betrayal, justice, loyalty, then math can’t stand for those things.
To say that math can express every aspect of reality that language can, is either to say that math is not limited to whatever has quantity or can be measured, or to say that reality is limited to whatever has quantity or can be measured.
As for saying we don’t speak in “math-language,” well, isn’t an underlying premise of Mark’s argument a distinction between the two, math and language?
Tony, the English word “justice” does a pretty terrible job of standing for justice. It’s clear that most of those who use the word have no clue what they’re talking about, and if there are any exceptions, they are exceptions by virtue of having at best a very slender clue, not by virtue of having anything like a complete understanding. Among the best of the feeble attempts at really getting a grasp on justice would be those from people like Plato (who seems to have thought all knowledge was ultimately knowledge of something like geometric forms), or Mill (whose utilitarianism treats all moral issues as depending on measurable effects on happiness), or Kant (who treats all moral issues as logical issues, and so as a kind of mathematical issues). Pardon me if I don’t immediately accept the claim of someone who doesn’t seem to understand mathematics at all that all of those efforts were utterly misguided.
Tony –
That is a large question, and like you said, I don’t think I can do justice to it here. I’ve been reading a lot of Cassirer lately, which is why I’m thinking about the topic of language now. He describes language as our initial attempt to halt the flux of experience, and begin to make it comprehensible; he alludes in one place to a Babylonian creation myth where the primordial chaos is called a time “when nothing has a name.” Naming is for him the way we first make the world present to our minds in cognizable fashion. But he also emphasizes that this is not a passive process, that we actively shape our world through our language (he’s a Kantian), and the world that we create most first and foremost is the community, the world of inter-subjective relationships. So language becomes a key way of mediating our dealings with one another. I’m not even beginning to do Cassirer’s work justice, but hopefully I’m making the point that he has some very compelling reflections that touch on your question. He has a great little book called “Language and Myth” that contains a lot of these reflections.
Aaron,
I guess if Plato, Mill and Kant were only feeble attempts at really getting a grasp on justice, then there’s really no need to discuss them, is there?
Based on your comment about Kant, it seems that you reject the position that math is limited to whatever has quantity or can be measured, and instead equate mathematics with logic? Or any logic-based system of symbols? I’m not trying to put words in your mouth, just trying to understand your argument.
Can I interject briefly?
I think maybe a lot of this (very valuable) discussion could be cleared up by a few essays by a guy that’s held in high esteem by this site: Walker Percy.
“The Delta Factor”
“The Message in the Bottle”
“Metaphor As Mistake”
all of these can be found in the essential volume “The Message In The Bottle”
In them he makes use of Charles Peirce’s ruminations on the ‘Semantic Triangle’ as the best theory we have for understanding language. His basic point is that language has a tripartite representation of reality, whereas math (read: science) has a necessarily dyadic upon which is layered various levels of complexity. Language, he argues, is irreducible complexity, a tripartite relationship between sign (uttered word), and referent (thing) and you, the one that makes the association. He points out a ghostly ‘third thing’ that happens. As you associate an arbitrary phoneme: ‘a-p-p-l-e’ to a thing you experience: ’round, sweet thing you can safely eat’ a third thing is created in your mind, the symbol, the apple as such that is neither sign nor referent, but is at once both and more. It’s how we can talk abstractly instead of just talking about specific things that we relate causally.
He proves this by asking you to say a common word over and over until it sounds ‘weird’. That is the symbolic weight of the word being stripped away through repetition until the word just becomes a naked phoneme and sounds as unfamiliar as 694812. It’s your mind that forms the connection.
So, I think what is being argued above about a ‘mathematical representation of reality’ is only being talked about as “can numbers stand in for names and plus, minus and radical symbols stand in for verbs. We can certainly conceive of a strange ‘future-language’ where all our words are numbers, but I think Percy would say that in order for that to happen, we would have to associate numbers as symbols rather than quantifiable bits of information, so they would cease to be true numbers and their relationships would cease to be math.
Tony, there is no meaningful distinction between logic and mathematics. I do not know if logic can deal with anything that is not measurable, as I am not the one who introduced this word “measurable” into the discussion, and I find it rather hopelessly unclear. But if there are unmeasurable things which are graspable by logic, then it would follow that mathematics can deal with some unmeasurable things (those things). Similarly, if the uncountable is always unmeasurable (I don’t know if it is; again, I find “measurable” hopelessly unclear), then that would provide endless further examples of mathematics dealing with the unmeasurable.
And I suppose if you think that you and Mark have a non-feeble understanding of justice, this wouldn’t surprise me in light of the pattern you display on the subject of mathematics, of being overconfident in proportion to your ignorance. But don’t beat around the bush; are you really claiming that you understand justice much better than Plato, Kant, or Mill did?
Mr. Boyden – You’re wrong. Logic and mathematics are not identical. Philosophers like Frege and Russell tried very hard to demonstrate that they were identical, and were never able to do so. Nor would it have the least bearing on my argument if they were identical, since logic is equally inapposite to the whole range of human experience. Formal systems like logic and mathematics are, by definition, highly abstract modes of reasoning which cannot capture the full specificity of human experience. This is something which most intelligent people realize, but if you have demonstrated any ability here in this discussion, it’s the ability to miss what most intelligent people realize.
Mark, you don’t even know who Alonzo Church is, and you’re trying to tell me about the relationship between logic and mathematics? Russell has been dead for more than 40 years, and didn’t contribute much to logic for at least 40 years before that. His failures are hardly relevant to the situation these days. The merits of the neo-logicist project (you probably don’t know there’s such a thing as neo-logicism, do you?) are controversial, but mostly not because nobody thinks it can succeed. It may in fact already have succeeded; it’s hard to tell, because the criteria for success are so unclear. But it’s not taken nearly as seriously as Frege and Russell took logicism, not because people think logic is something separate from mathematics, but because they think the whole idea that mathematics needs a specific foundation is mistaken. So even if the neo-logicists can derive all of the rest of mathematics from something they call logic, most these days would just yawn at yet another relative consistency proof.
If it wasn’t already obvious for other reasons, the inseparability of logic and mathematics is quite clear from Goedel’s work; all of what are generally regarded as logical structures can be modeled within mathematical structures (provably; Goedel proved things, he didn’t traffic in the kind of hand-waving you do). Since pure logic is only about structures, and pure mathematics is only about structures, it doesn’t make any sense to say they’re talking about different things when they’re talking about the same structures.
It may have succeeded, huh? Well then, I may stand corrected.
It doesn’t matter whether neo-logicism has “succeeded” or not. Logic and mathematics are not distinct regardless; the only question is whether logic is a part of mathematics or the whole of mathematics (and the reason I put the quotes around “succeeded” is that as I already mentioned, most modern logicians and mathematicians don’t think that question really matters very much).
“Not distinct from” generally means “identical to,” not “a part of.” In that sense, the stripes are distinct from the zebra; Extremadura is distinct from Spain. Your claim that “logic and mathematics are not distinct,” if taken as those words are usually taken – ie, that logic is identical to mathematics – is not true; as your own words make plain, such a thing has never been demonstrably proven (and it doesn’t matter whether anyone thinks its worthwhile to prove it or not). Anyway, none of this has anything to do with my thesis since, as I noted, logic is as alien to human experience as mathematics. Logic is not language, nor is it anymore capable of representing the full range of experience, as language is. A translation of the “Ode to a Nightingale” into symbolic logic would be as quixotic an endeavor as a translation into mathematics.
As a matter of fact, this conversation has gone way off the rails, due to your attempts and the attempts of the Illiterate (formerly, Rusty) Philosopher to throw out one trite, immaterial objection after another to a point that most normal people can grasp instantly. Numbers can be made to stand for letters, computers can be programmed to recognize bad syntax, words are often imprecise in their denotation – these are not serious objections to the point I was making and, frankly, if that’s the best you can come up with, I think my argument holds up pretty well. I will reiterate the main point of my article once again – science began with the attempt to delimit its field of inquiry, in order to produce more conclusive results than had hitherto been produced. It is now claiming that its field of inquiry is unlimited. This must either mean that it has abandoned its original methodology, in which case we have great cause to doubt the conclusiveness of its findings, or there is something dishonest going on in the scientific world today. Nothing you or your dopey friend have written would call into question this thesis for a minute. And since you are both determined not to understand anything I have written, continuing this conversation would obviously be futile.
Actually, I can’t leave this conversation without sharing the following quote from Ernst Cassirer, making very nearly the same point I’ve been making all along: “The true ideality of language exists only in its subjectivity. Hence it was and always will be futile to attempt to exchange words in the various languages for universally valid signs such as mathematics possesses in its lines, numbers, and algebraic symbols. For such a system would express only a small part of a what can be thought, it would serve only to designate such concepts as are formed by purely rational construction. But the substance of inner perception and sensation can be stamped into concepts only by man’s individual representative faculty, and that is inseparable from his language.”
I’m waiting for Mr. Boyden and the Rusty Philosopher to come on here and tell us that Cassirer, one of the most learned men of the twentieth century, was also clueless, uneducated, etc.
Father of Quantum Mechanics:
“The physicist may be satisfied when he has a mathematical scheme and knows how to use it for the interpretation of experiments. But he has to speak about his results also to nonphysicists who will not be satisfied unless some explanation is given in plain language, understandable to everybody. Even for the physicist the description in plain language will be a criterion of the degree of understanding that has been reached. To what extent is such a description at all possible? This is a problem of language as much as of physics…. (168)
Furthermore, one of the most important features of the development and the analysis of modern physics is the experience that the concepts of natural language, vaguely defined as they are, seem to be more stable in the expansion of knowledge than the precise terms of scientific language, derived as an idealization from only limited groups of phenomena. This is in fact not surprising since the concepts of natural language are formed by the immediate connection with reality; they represent reality. It is true that they are not very well defined and may therefore also undergo changes in the course of centuries, just as reality itself did, but they never lose the immediate connection with reality. On the other hand, the scientific concepts are idealizations; they are derived from experience obtained by refined experimental tools and are precisely defined through axioms and definitions. Only through these precise definitions is it possible to connect the concepts with a mathematical scheme and to derive mathematically the infinite variety of possible phenomena in this field. But through this process of idealization and precise definition the immediate connection with reality is lost.(200)
We know that any understanding must be based finally upon the natural language because it is only there that we can be certain to touch reality, and hence we must be skeptical about any skepticism with regard to this natural language and its essential concepts. (201-2)”
Werner Heisenberg, Physics and Philosophy: The Revolution in Modern Science (New York: Harper & Row Publishers, 1958).
I’m a bit late to the party here… but I’d like to point out what seemed obvious me from reading your first few paragraphs: your claims of the degree to which our experience is conditioned or even determined by language are rather overstated. I will grant that language influences our experience, sure. But to state that such fundamental emotions such as indignation, gratitude and shame would not even _exist_ were it not for the language that represents them (end of 2nd paragraph) seems to be quite an exaggeration.
Which came first, after all? The emotion or the vocabulary that describes it? This is not a chicken-or-egg question. Isn’t it obvious that the emotion had to pre-exist the vocabulary? We do not have words for emotions we do not feel. You would counter, of course, that we do not have emotions for which we do not have words. That is an unfalsifiable claim, however, since if there were an emotional state that I could not describe by language, it cannot be produced as evidence in this linguistic conversation we are having. That doesn’t necessarily mean that such emotions do not exist, however.
The empirical evidence I bring is non-human primates. It is well-known that chimpanzees, for example, exhibit a wide range of “human” emotions: justice, shame, indignation, affection, etc. Emotions are as much a fundamental part of the day-to-day existence of a chimp as they are for us, as far as we can tell. Yet their language is far from human. There is no evidence that chimpanzees communicate with the degree of symbolic representation that we do. Ergo, emotions and experiences are not determined by language.
I’m not making the claim that your thesis is fundamentally flawed. I agree that language does mediate and influence our experience, and so if mathematics cannot represent that mediation and influence, it cannot fully represent the human experience. I just think your case is somewhat overstated.
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