Dühring expresses something I’ve run into before (Heinlein pulls a slight-of-hand with it in Rocket Ship Galileo; I noticed the slight-of-hand when I was 12). It is the belief that mathematics can be produced “without making use of the experience offered us by the external world.” In other words, that mathematics is it’s own thing, fully abstract, existing only in the mind, and independent of the content of the world–that it exists in and for itself. The validity of mathematics is mathematics; there’s no connection to anything else.
Engels concedes that pure mathematics is independently valid from the particular experience of each individual, but then, so is every other fact. But, “The concepts of number and figure have not been derived from any source other than the world of reality. The ten fingers on which men learnt to count, that is,to perform the first arithmetical operation, are anything but a free creation of the mind.”
I want to take a moment here for something that is probably a side note. The subjectivists, the schematists, the empiricists–that is, generally, those who are hostile to theory–nearly always have this in common: an impatience for and a dislike of history. This seems to hold true whether the subject is social, political, philosophical, religious, or about the natural sciences. While they may be willing to talk about what is (generally in a fixed, static, lifeless way), the process by which anything became the way it is, is contemptuously set aside as unimportant. And yet everything in the world is constantly in movement, everything is part of a process. To blind one’s self to history (in this case, to ignore or disregard that mathematics came from arithmetic, which came from attempts to understand nature, from abstracting number from real world objects) can only produce distorted, or, at best, severely limited, narrow, one-sided understanding.
“Counting requires not only objects that can be counted, but also the ability to exclude all properties of the objects considered except their number — and this ability is the product of a long historical development based on experience.”
“Before one came upon the idea of deducing the form of a cylinder from the rotation of a rectangle about one of its sides, a number of real rectangles and cylinders,
however imperfect in form, must have been examined.”
“Like all other sciences, mathematics arose out of the needs of men: from the measurement of land and the content of vessels, from the computation of time and from mechanics.”
Here, too, by the way (although Engels doesn’t yet go into it) we have a dialectical relationship. We study triangles in life, and produce the abstraction called trigonometry , we then use trigonometry to both design and use sails to drive a square-rigged ship, and then we use that experience to further refine our understanding of trigonometry. We have the mathematical concepts abstracted from nature, we have those concepts used in order to further understand aspects of nature.
In the end, the theory has built upon itself to the point where the impressionist is able to argue (albeit unconvincingly) that mathematics is just about itself. The fact that we are constantly using higher and higher forms of mathematics in our understanding of nature; and in technology to change and examine nature, must be, to these people, pure coincidence. And that these discoveries, in turn, are folded into mathematics and spur it still higher, they simply prefer not to see.
17 thoughts on “Anti-Dühring Part 5:Chapter 3 (Concluded)”
It’s not yet 9p, so here goes…
“The subjectivists, the schematists, the empiricists – that is, generally, those who are hostile to theory – nearly always have this in common: an impatience for and a dislike of history…while they may be willing to talk about what is (generally in a fixed, static, lifeless way), the process by which anything became the way it is, is contemptuously set aside as unimportant.”
My experience is just the opposite. I’ve generally found people who like to theorize about politics, philosophy et al. (in the general, non-scientific sense of the word) tend to be solipsists. Their first-hand experience is all that matters to them. They don’t look at the big picture, often they don’t even acknowledge there may be another point of view, much less historical facts of which they’re unaware.
This is probably a case of confirmation bias (on both our parts). I’m not keen on talks about political theory and changing society unless it leads to concrete ideas and so I remember the bits that annoy me when people babble without making a point.
(There’s a subset of that type of person, people so taken in by the minutiae of history that one can drown in the pointless details they keep bringing up, details they can never tie in to anything pertinent to their topic. I’m really biased against that type.)
Steven, you have already pointed out that you have no obligation to argue about such things, and truly it is something where different opinions are defensible.
The teachers and mentors for pure math during my undergraduate and graduate work all seemed to believe that they were doing math for math’s sake, and that their work was independent of the outside world. And yet when they wrote grant applications they tended to stress possible real-world applications. Like, they would argue that physicists had found uses for some results from their areas. (Typically when I looked at what that meant, there would be some obvious 3D geometry conclusion that anybody ought to see was true just by looking at it, but physicists had in fact gone 20+ years without noticing it. Then the first guy who did notice would find the most abstract theoretical result he could to quote as proof, usually something that worked in any number of dimensions. The math was used only for the proof, and then the physicists would go back to business as usual.)
So this is a sort of argument by authority. But just because real-world pure mathematicians believe my way does not mean they are right. They might not understand the historical process.
Yes, math developed first to solve practical problems in the real world. It only went further when there were resources available to support it. Egyptian geometry came from surveying, and there was probably some pure geometry practiced by secret societies among priests who had the farmers’ surplus and leisure. Nonsecret abstract geometry was done by upper-class greeks who had slaves to give them leisure. I don’t know who did pure math in ancient china or india etc. We have had pure math and attempts to find proofs and solutions for impractical problems about as long as we’ve had writing. But if it got done by poor people we don’t have surviving writing from them.
However, consider the history of art. I don’t know a lot about art, but I know that historically art was always representational. Art is about making models of the real world, and nonrepresentational art is all very recent and many people think it is mostly worthless. Also, throughout most of its history, art was commercial art. The idea of “art for its own sake” is recent and not particularly significant. It only matters if you think those particular artists matter, or their work. Probably a similar argument could be made about pure math.
It made sense for Engels to draw his conclusions from what he knew then. His arguments are kind of bogus, but his conclusions could be correct anyway. It is clear that Duhrens was wrong. Duhrens thought he could create mathematics from scratch with no reference to the real world, and use it to find truths about the real world from scratch. But if you start from scratch, how do you choose which axioms will fit the real world? You can’t tell which versions of math will fit until you check with the real world to find out. Even if it is true that mathematics does not have to be derived from the real world, it’s only the particular math which can be derived from the real world which will fit the real world. To derive the real world from pure math you have to cheat.
And even then at best you get partial results. Some pure math kind of fits the real world, because you chose the kind of math that would fit. But even then, it only fits some abstractions, the particular details you choose to keep when you compare to the math. People could assume that if they used *more* math they could fit everything at once with no abstractions. They could take that on faith if they wanted to, but there is no particular reason to believe it beyond sheer faith. Probably it would take an infinite number of axioms, and human beings could never come close to doing it.
L. Raymond: I’m just a little confused. We seem to be speaking of different things. That is, I’m saying groups A, B, and C show characteristic X, and I hear you saying, “No, group D shows characteristic Y.” What am I missing?
Mathematics can be divided into two very large fields: applied and theoretical. Applied math as its name implies, is that which has direct application to the real world. Within theoretical, there are some fields that also can be reduced to real world scenarios and then be useful. And there have been times that incredibly obscure stuff has been found to be of use in physics, etc.
However, I have to say that a great deal of theoretical math is in fact done for its own sake….to give a very basic example, perhaps someday knowing whether or not c = aleph (1) (sorry, don’t know how to do special characters or subscripts) will be useful, but for now, I think I can safely say that such endeavors exist only in the mind and have no particular relevance to the real world.
I myself studied mathematical logic, specifically set theory and model theory….some of which is applicable to real stuff outside of one’s mind, but mostly not. The point I’m after here is that there is a fair bit more to mathematics than counting, designing sails and explaining nature. It does all those things, but also quite a bit more, much of which will never see the light of day outside of the hallways of a university.
J Thomas–I’d hate to say “all” but I’m sure most mathematicians (if not in fact all) know how their field evolved. We do learn some history of math as we go along.
Looks like three times now in the last couple of days I’ve ventured into the fray here from my normal lurking. Hope you don’t mind….it seems you all know each other, so I hope I’m not intruding. As a means of introducing myself, I’ll say that I can be blunt, but I try to calm that down so as not to sound like too much of a jerk. Apologies in advance if I forget. Also sorry this is a bit long. I don’t normally talk so much.
KellyM: We do not, for the most part, know each other. You are most welcome. So, then, there is a hard line between theoretical and applied and they do not transform into one another, or feed into each other, and anything that starts in one category stays there, is that correct? :)Fe
I was not and am not denying the usefulness (or existence!) of “pure theory” in mathematics or anywhere else; I am denying that it has an origin and existence utterly and forever apart from the real world. Any branch of knowledge (even something as emphatically real-world based as geology) will produce theories that then build on each themselves and each other. That is insufficient ground to dispute the point that the real world is the origin, and ultimately (even if not today in a given case), the testing ground. Am I wrong?
There is some wiggle room. That is, there aspects of pure mathematics that are useful to the applied mathematicians, and there are fields of the pure math that end up, either by design or by luck or accident to be useful one day, and there are things like linear algebra, which is very useful both in the pure and applied worlds. But, that said, there is an awful lot of pure math that I really can’t see being “useful” any time soon….
To be fair, the distinction between pure and applied used to be a lot fuzzier, but the field has changed a lot in the last 100 years or so, and mathematicians now *really* specialize, and most tend to choose a very narrow field in which to do the bulk of their research. So for a lot of them, yes, there is a very hard line between theoretical and applied, and I can say that the theoretical has truly grown into its own thing. I think the guys from the 19th century (especially non-mathematicians) would not recognize the field today. There’s even something called “category theory” that studies the various structures of math and generalizes things even further. We used to marvel in grad school that we had professors, all PhDs in math, that really had very little they could “talk shop” about, as their specialties were so different. That said, my knowledge is on the pure side of things. I wasn’t so interested in the applied stuff. Too messy for my tastes. :)
Oh, and to be clear, I have a master’s in math, not a PhD, though I did toy with the idea of getting one. In the end, the master’s was enough to go into my chosen field at the time. Just wanted to clarify that, in case my blathering gave a different impression.
Whoops…as I was writing you updated your post…..I’ll comment on it in a bit….real life is calling at the moment. :)
KellyM, did you read the Engels chapter?
Engels was arguing against a stupid philosopher who got everything wrong. But in the bigger historical context, he was taking a stand that looks basicly correct today. For example he argued against Vitalism, a theory that said there is something fundamentally different about life which makes it and its products completely different from the nonliving world. It was widely considered a big blow against vitalism when a german chemist figured out how to create urea, a chemical of life, from completely nonliving reactants.
The opponents were looking for ways to transcend the physical world. They thought life was special. They thought thinking was special. They thought human thinking was more special than animal thinking. They thought mathematics was more special than other human thinking. They kept looking for things that did not come from the material world, because they wanted there to be something more and separate.
But materialists said it all came from the material world and there was nothing more.
It should be obvious that in 1878 this was not something that could be decided by experiment. Wouldn’t it make sense to leave the question open until science could find answers? But somehow the one group insisted that there had to be something more, and then for an antithesis the other group insisted that no there definitely was nothing more. Both sides used arguments that are fundamentally bogus.
Engels said that even though math can be created out of sheer thought, still in reality the history was that it came from nature. It was created by natural human minds, and it was created to fit nature, to meet practical needs. So the theorists who say things like “God gave us the natural numbers, all else is the work of man” are irrelevant. In a material world, all of mathematics must be found in nature, and all the methods we use to discover math must come from nature. Our thoughts come from nature and are always congruent with nature. We are incapable of unreal thoughts because if we could think things that do not fit the natural world, materialism would be wrong. And it is not. Or something like that.
I think we now have enough data to make a fine synthesis. Materialism fits our experience provided it does not get stretched out of shape. Life and thought and human thought all display emergent properties but they do not violate the laws of physics etc as we know those laws today. (If we do find something that violates those laws, we will change the laws enough so they won’t be violated.) Everything we know about is compatible with the material world, and there are special things that in addition follow their own rules.
L. Raymond – I have the same confusion skzb does.
KellyM – The tv show Numb3rs has convinced me that all math, no matter how theoretical, can be used to solve crime. (Also, hi! Also also, a couple of people DO know each other IRL but it seems to me like a bunch of the commenters around here are just comfortable enough that it seems like everyone knows each other.)
JThomas – “did you read the Engels chapter?” Which chapter?
And on to my comments. I continue to like how every passage I highlight in my copy has already been copied by you in your comments. Although you are possibly getting less comment out of me because of that.
You quoted the following line: “Like all other sciences, mathematics arose out of the needs of men: from the measurement of land and the content of vessels, from the computation of time and from mechanics.”
…but I think it is better if you include the sentence immediately following: “But, as in every department of thought, at a certain stage of development the laws, which were abstracted from the real world, become divorced from the real world, and are set up against it as something independent, as laws coming from outside, to which the world has to conform.”
This is important to me because it says that the higher levels of math (the more theoretical, not applied?) – or anything else – are formed in such a way that you can’t see the fingerprints of the so-called real world on them anymore. Even if there aren’t visible artifacts of how a thought or natural law was abstracted from the real world, that’s still how we got it.
The most important bit of this entire chapter, though is where he writes: “it is fine that Jen doesn’t understand math past trigonometry; she won’t need it to grasp the rest of this book.”
“This is important to me because it says that the higher levels of math (the more theoretical, not applied?) – or anything else – are formed in such a way that you can’t see the fingerprints of the so-called real world on them anymore.Even if there aren’t visible artifacts of how a thought or natural law was abstracted from the real world, that’s still how we got it.”
I think that’s exactly right. And, yeah, it’s valuable to emphasize that this is what always happens with a scientific discipline at a certain point in its development. If I knew a little more of the history of science (I know snippets, which I find fascinating and wonderful), I could probably draw conclusions about when that happens.
But the development of a scientific discipline is a process; it is easy to look at where that discipline is now (in terms of methods, approaches, means of acquiring new data and theories, &c) and believe that that is simply how it works. No, that is how it works now.. A hundred years ago it was different, and there is no reason to believe that a hundred years from now it won’t be different again. Not only does human knowledge not stand still, but our approaches to discovering human knowledge don’t stand still, either within or among what we call scientific disciplines.
Marxism is the science of the struggle for the emancipation of the working class. But anyone who thinks it’s been standing still is woefully ignorant.
The most important bit of this entire chapter, though, is where he writes: “it is fine that Jen doesn’t understand math past trigonometry; she won’t need it to grasp the rest of this book.”
Huh. I read that wrong. I thought it said “Steve” instead of “Jen” and “geometry” instead of “trigonometry.”
@skzb: That is, I’m saying groups A, B, and C show characteristic X, and I hear you saying, “No, group D shows characteristic Y.” What am I missing?
I doubt you’re missing anything. You’re saying “they nearly always do X”, and I’m saying “I find the opposite to be true, but I recognize my own bias”, and I went on from there. In other words, I disagree with your characterization of people who tend to dislike pure theory.
I didn’t read all the previous A-D posts, so it’s posible you’re using “theory” in a manner specific to this series, but I interpreted the phrase “hostile to theory” as meaning people who don’t enjoy talking about *why* things are a certain way unless it’s accompanied by a concrete resolution: “and here’s what we will do to change it”. Not “should” do, or “could” do, but will do. And if the speaker has no intention of doing anything at all, I often wonder why she’s bothering me in the first place.
As I recall you’ve said I’m an empiricist. I am unquestionably bored by people who drone on and on about why they think X has happened or why situation Y exists, but it’s because such people are invariably myopic; they interpret everything through their own bias without acknowledging it. For instance, if they’re die-hard anarcho-capitalists, they interpret everything in economic terms and theorize every step of human history was as a result of stifling or glorious economic policy, depending on whether they approved of what came next. This is why I said I tend to find theorists to be solipsicts – their biases and their desires determine what theories they hold, and they never seem to feel the need to test their theories via practical action. Granted solipcist may not be the right word, but the only other phrase that comes to mind is so negative I don’t feel right using it.
On a different note, after this series have you any intention of going over “Herr Vogt”? It seems to me they’re the same in both being directed at specific people and their ideas, although I admit I never did get far through “HV” and am not sure if that holds true for the whole work.
skzb: Sorry I didn’t get back last night. After dealing with various stuff, sleep was necessary. So, to discuss the second bit of your comment, as I hoped to do last night…
As for origins, well, I certainly can’t argue that math hasn’t arisen out of real world issues. What I am saying is that there is a lot of math that has arguably gone way beyond that, and unless we can find infinite set of various sizes and structures, a lot of say set theory and logic will remain forever an intellectual endeavor, and not be testable in the real world. Unless of course it turns out that we are inhabitants of just one universe of infinitely many and we find a way to traverse them all, and then can use that somehow as a testing ground…who knows. :) That would be really cool, and I would go back to get that Ph.D. to explore stuff if that turned out to be true.
But, scifi possibilities aside, I will still maintain that there are things in pure mathematics that cannot be tested in the real world. I mean, we have different axiom systems in set theory that are all perfectly valid that contradict each other. At various times someone doing research may work in one or the other system, for various reasons. Even if we can play with one of these systems somehow in the “real world” to try out something, we clearly then can’t play with the others that contradict the first one. I’d also love to see someone try to test out some of the wackier-sounding things you can prove from the axiom of choice, but I wouldn’t hold my breath.
I like what jenphalian said about things developing to a point that you can no longer see the fingerprints of their origins anymore….that sort of gets to what I was thinking in regards to math. Yes, it comes from human minds, and as such is “natural” but it no longer has to represent anything *in nature.* None of that probably affects any of the other discussions though.
Honestly, I was only drawn into the discussion of this book because of the references to mathematics. I’m less likely to follow along with the rest of this book, but I’m glad my butting in on this bit was OK. In fact, after this, I may get back to reading a history of math book that I had started and put down a year or so ago when life got too busy to think about it…..so thanks for that. I’ll still read your *blog* though, so perhaps I’ll less shy about butting in again on something else later. :)
jenphalian: Well of course it can, math can do everything! I’ve actually never watched that show. I probably should.
J Thomas: Yes I did read the chapter, but I was mostly responding to skzb’s comments on the material…specifically to the stuff on math, because as stated above, that’s my thing, though I’ll confess back in grad school, I was less interested in the philosophy of math and the philosophy of logic than I was in the math and logic themselves.
L. Raymond: I use “theory” to mean generalized experience.
“And, yeah, it’s valuable to emphasize that this is what always happens with a scientific discipline at a certain point in its development. If I knew a little more of the history of science (I know snippets, which I find fascinating and wonderful), I could probably draw conclusions about when that happens.”
I have some examples. Like, evolution. Darwin presented a whole lot of evidence that it happened in nature. His theory about how it happened was weak, and made some implausible assumptions because he couldn’t think of better ones.
Then Mendelian genetics got applied, and suddenly it was all workable. Around 1910-1940 they worked out the math to describe it, and the math worked just fine. At that point they were ready to apply the math anywhere it might fit. There were theoretical evolution studies, looking at ways that genetic systems could be organized to make evolution work faster, etc. No immediate relation to data, strictly theoretical. Starting around 1965 there were attempts to apply evolution theory to computing. Solve multi-dimensional math problems with repeated guessing that fit the evolutionary methods. “Genetic algorithms”. Conditions for evolution in computer programs did not need to have any relationship to Mendelism, they could do anything they thought would work. They tried a big variety of methods, many that had no inspiration at all from biological evolution. And by this time they had some precise definitions. Evolution will occur with anything that does imprecise reproduction, with selection of some variants over others. That applies to life, but it also applies to a lot of other things unless you use that as a definition for life.
It started with a description of nature. Then they managed to abstract out the rules for the special things they were looking at. Then they applied those rules to anything else they could fit them to. After that they generalized the rules so they could apply them to even more different things. The generalized rules no longer have anything in particular to do with nature, but some specialized subsets of them do.
“But the development of a scientific discipline is a process; it is easy to look at where that discipline is now (in terms of methods, approaches, means of acquiring new data and theories, &c) and believe that that is simply how it works. No, that is how it works now.. A hundred years ago it was different, and there is no reason to believe that a hundred years from now it won’t be different again.”
Yes! That is centrally important.
“Marxism is the science of the struggle for the emancipation of the working class. But anyone who thinks it’s been standing still is woefully ignorant.”
And that. I run into fundamentalist Christians who read Darwin with great concentration, because they think he is the founder of evolution so they regard his writing as its most important scriptures. When they think they find an error in his work then they think that all of evolutionary theory is about to topple.
If it were to turn out that Marx or Engels occasionally made a mistaken argument or jumped to a conclusion, that says nothing about modern Marxism.
I agree with Steve (and Engels here) that mathematics arose out of a study of real world objects. The logical consistencies of mathematics can be viewed and extended to objects that don’t exist (or at least are unobserved) in this universe and that is a cool and useful thing.
I certainly agree that the scientific and mathematical disciplines are processes that constantly improve upon (and invent new) the works of the past. Duhring’s assertion of an absolute system is pretty foolish at all and even more foolish in that the particular system he was asserting was so very flawed in its foundations.
“Huh. I read that wrong. I thought it said “Steve” instead of “Jen” and “geometry” instead of “trigonometry.””
Well, we’re reading different editions, after all.
KellyM – I don’t know how good the math in Numbers is – it might be wrong or simplified wrong, in which case it may drive you crazy, but it is totes a fun show.
“Marxism is the science of the struggle for the emancipation of the working class. But anyone who thinks it’s been standing still is woefully ignorant.”
“But, scifi possibilities aside, I will still maintain that there are things in pure mathematics that cannot be tested in the real world. I mean, we have different axiom systems in set theory that are all perfectly valid that contradict each other.”
KellyM, yes! It’s hard to get this across to people who haven’t experienced it.
So for example, we have some axioms for the euclidean plane. If one of them is different, you get the surface of a sphere. Make it different a different way and you get a hyperbolic space.
Another different way and you get a cross-cap.
Or a klein bottle.
A variety of other shapes have been studied.
People used to think the surface of the earth was an approximation to a euclidean plane. Now we know that it is an approximation to the surface of a sphere. But what about the whole space? We used to think that space was a euclidean 3-space. Einstein hypothesized that it was actually a Minkowski space due to peculiarities in the way light travels. He said that if you shine a light in all directions from a particular place at a particular moment, then whenever the light reaches some other place, the Minkowski distance between that time and place versus the original, is zero. To me that does not say it is really a Minkowski space….
There are a variety of 3D or 4D spaces that are each internally consistent, that we could in fact be living in. We like euclidean 3-space because it is easy to think about. But to test whether we actually had something else, we would have to measure from very long distances, and so far we can’t do that. We can only look at light which has traveled from long distances, and some cosmic rays and neutrinos. We don’t actually know the shape of space.
We have gone from assuming that Euclid applied to the real world, to mostly assuming that Euclid applied to the real world despite having many other candidates. Plus we have a lot of consistent models which do not apply to the real world unless we find some novel way to apply them.
Engels’ historical claims are about right. We went from practical problems to abstract geometry sometime around the invention of writing. Then we had one single geometry that was supposed to apply to the world, for around 2400 years. Knowledge of geometry varied, but still there was only one. We have had alternative geometries for less than 200 years, and now nobody really knows which of them might apply to the real world.
“A man with one clock knows what time it is. A man with multiple clocks is never sure.”
What might show up within the next 100 years?