This whole chapter gives me the headache. For one thing, I had more trouble here than usual in separating Dühring, Hegel, and Engels. I wish he’d used more direct quotes, or been more precise about when he was summarizing. But even aside from that, contemplating infinity hurts, and, before I attempt it, I want some reason to believe it will be useful. I mean, I accept that Herr Dühring said some idiotic things about it, but that isn’t the point of reading this book. Do Engels’ considerations about infinity of time and space actually produce any positive knowledge? Let’s see.

Okay, we’re mostly dealing with absurd definitions of infinity, and attempts to treat it as if it were finite, with predictable results.

And then he quotes Kant, about whom the most that can be said is that he is more easily comprehended than Hegel. But then, so is Cervantes in the original Spanish, and I don’t speak Spanish.

Alright, that next part I sort of get. There is a distinction between an infinite series–which starts at 1 and continues forever; and infinity in space, which has no starting point in any direction. Gotcha. Now onto time:

“But if we think of time as a series counted from *one* forward, or as a line starting from a *definite point*, we imply in advance that time has a beginning: we put forward as a premise precisely what we are to prove.”

“For that matter, Herr Dühring will never succeed in conceiving real infinity without contradiction. Infinity is a contradiction, and is full of contradictions. From the outset it is a contradiction that an infinity is composed of nothing but finites, and yet this is the case.”

“It is just *because* infinity is a contradiction that it is an infinite process, unrolling endlessly in time and in space. The removal of the contradiction would be the end of infinity.”

Okay, yeah. Infinity is itself a contradiction–that is, simultaneously one thing and its opposite (I know there are those who do not believe contradictions are possible in nature; we will deal with them in due course).

“For the basic forms of all being are space and time, and being out of time is just as gross an absurdity as being out of space.”

In essence, Herr Dühring is suggesting a state previous to time, and thus previous to motion. With the entire world in such a state, whence would come the initial motion? Answer: from outside the world, or, in other words, God. Therefore, if we are not to become hopelessly tangled in absurd contradictions–or bring in God by the back door–we cannot assign a beginning to time apart from matter. Time exists as the motion of matter; matter moves through in time. Engels does not actually *say* that time is a fourth dimension, but he sniffs around it a bit, which is fairly impressive in the 1870s.

Really, I didn’t find a lot in this chapter that did more than crush Herr Dühring. This by itself may give us a clue as to the importance of infinity to Engels’ worldview.

I say that infinity is a concept that is sometimes useful, and we do not encounter any infinities in nature.

Try a simple version. We have the concept of splitting something into three equal parts. But we want to think about all our fractions in decimal, because we like it that way. So we don’t just want to write down the number 1/3.

3/10 is an approximation, but it is wrong.

33/100 comes closer, but it is still wrong.

333/1000 is closer still.

If we divide 1/3 into an infinite number of pieces, each of which can be expressed exactly in base 10, we can get something that exactly expresses 1/3. It takes an infinite number of pieces but we can do it. Further, we can do arithmetic on an infinite number of pieces if we follow certain rules. We don’t actually have to handle an infinite number of pieces, but we can make most of them cancel out and still get the right answer. So the result is useful. We can follow our arbitrary preference to express everything in base 10, and still get exact answers to some things that simply do not fit base 10 fractions, if we follow the correct rules for handling infinite sums.

This gets expanded into complicated rules for dealing with complicated infinite sums. It gets expanded into ways to use the biggest parts of an infinite sum and throw away the small ones, and still be accurate within known bounds. There has been a whole lot of work done on this sort of thing for practical purposes.

Does nature use these concepts? No. When nature divides something into three parts it just divides something into three parts. It does not sit down with an abacus and do infinite sums. It does not need to. We have a concept that works for us to model nature. It gets the right result even though we start out using whole numbers and fractions that don’t always fit nature. We have to go through various tortuous contortions to make it work, but it does work. There is no reason to think nature does it our way.

What about infinite distances? I dunno. I haven’t seen an infinite distance yet. Do you think there are infinite distances? Maybe you could go look and tell us what you found…. But if there aren’t infinite distances in the physical universe, what are there? How can it be anything other than an infinite distance? I dunno. Maybe you could go look and tell us what you found….

What about infinitesimal distances? I dunno. I haven’t seen those yet. We got microscopes but they can’t see less than the wavelength of light. We got electron microscopes and they can see smaller. Maybe someday we will get neutrino microscopes that can see smaller still. Current theory does not predict that, but who knows. Physics theory predicts that you can’t do anything with distances smaller than a Planck distance, that’s the smallest. Is that theory right? I dunno. Nobody has disproved it yet.

We have concepts of infinity that are sometimes useful in the real world because they patch up other concepts that do not fit the real world unless they are patched. We never actually see infinities in nature. Are they real? They are as real as other concepts.

“Infinity is a contradiction, and is full of contradictions. From the outset it is a contradiction that an infinity is composed of nothing but finites, and yet this is the case.”

I don’t see contradictions in current concepts of infinity. Mathematicians worked hard to avoid contradictions and I haven’t found mistakes in their efforts. This example is wrong by my understanding of “contradiction”. You can divide some infinities into pieces that are each finite. But you get an infinite number of such pieces. No contradiction. You can also divide some infinities into an infinite number of pieces, each of which is itself infinite. Engels was right to say that you can have infinities which are bounded in one direction but not in another, and also infinities that are not bounded in any direction. It goes every which way. Our universe could have a beginning and an end in time, or a beginning but no end, or no beginning and no end, or any combination. (Unless it turns out there is physical evidence that some combinations are in fact impossible. I haven’t seen such evidence.)

As usual, Engels demolishes Duhring’s position. In the process Engels says some things about the real world which might not be true. It’s hard to talk about the real world and be sure you have it right. He says some things about how human concepts have to work which definitely are wrong, but which probably do not matter much.

So I didn’t read anti-Duhring, but I thought I’d comment on the history of math here, since it seems like it might be at least tangentially relevant.

At this time, I believe that the prevailing trend of thought among mathematicians and mathematically minded philosophers was “finitist”: i.e. that infinity not only had no real existence in the world but was not even a legitimate value in mathematical theory. The very concept of infinity could not be tolerated by many famous thinkers throughout this period and even thereafter, including such varied luminaries as Poincare and Wittgenstein. As a transcendent quality, infinity could only be associated with God (if the philosopher was a theist or an idealist) and it would likely have had no place at all in serious discourse for many early materialists.

Of course there were mathematical approaches that made indirect use of infinity that had been worked in the West since the 17th century*, namely the infinitesimal calculus of Leibniz and Newton, and the infinite series of Taylor and others. But neither of those things contemplates infinity as a legitimate value of a formula or as a quantity in itself; instead infinity is used as a limit for integration or as a notional bound on the number of stages of a procedure, which will either result eventually in a definite finite value or in a undefined one (the dreadful horror of a singularity).

It wasn’t until Cantor that a formalized notion of infinity either as an ordinal (e.g. omega) or cardinal (e.g. aleph null) quantity entered into mathematics based on his new conception of set theory. Cantor’s first published work on infinity was in 1874, 3 years prior to anti-Duhring, but since neither Engels nor Duhring was a mathematician, I doubt either would have heard of it, and this is likely too because Cantor had to fight for a long time before his ideas became widely accepted; it probably wasn’t till Hilbert took up his ideas at the turn of the century that Cantor’s mathematics became respectable.

The point of this is that infinity as a notion, mathematical, physical, or conceptual, would I think have generally been regarded derisively at this time, and quite likely would have been considered by some socialist thinkers as a vestige or relic of some sort of idealist theistic sort of thinking from a past age. So possibly this may help with some connotations of the use of the word “infinity” at the time of the writing of the original screed.

*Indian math had a much more sophisticated and nuanced notion of different types of infinity that has some interesting parallels with Cantor’s cardinals, even before the common era, but no one in the West knew about it till later.

Miramon: Thanks. That’s very useful information.

Damn. I was writing something very similar to what Miramon just posted. Glad I hit “refresh” before I posted it….I hate being redundant.

My bottom line was that since the concept of infinite numbers was not well understood until after Cantor, really neither Engels nor Duhring understood it, and because of that, both made numerous logical mistakes in their arguments.

So, skzb, if this chapter gives you a headache, I wouldn’t worry about it. From the perspective of discussing “infinity” it’s really, um, not good. (trying to be polite. :) )

There is one other thing, regarding something J Thomas said above..since you ask about “infinities in nature” I would say that yes, we *do* see infinities in nature, since you seem to define an infinitely long decimal as an example of such…pi is a number that occurs in nature.

KellyM, this is a sort of philosophical distinction, but I would say it is possible to think that when pi is a number that would take us an infinite number of digits to describe, still nature does not use an infinite number of digits to describe pi. Nature just does stuff.

The ratio of the diameter to the circumference of an exact perfect circle is pi. There may not be any exact perfect circles in nature, and if there are they have a ratio. We cannot describe that ratio perfectly. We invoke infinity in our attempt to describe the ratio. But nature just has a ratio.

This is no problem if you use analog computing. The slide rule has pi marked on it, as accurately as any of the other numbers are marked. Just another number.

I believe it’s possible to interpret each use of infinities as the result of some flaw in our concepts. The concepts require an infinite number of fixes to get conceptually exact. This does not imply infinities in nature. If space is continuous, we would need infinities to describe it exactly. But space does not need infinities to describe itself, and perhaps it is not in fact continuous.

I don’t claim that a concept of nature without infinities is correct. Maybe none of our concepts are correct yet.

If, by some wild chance, you are interested in pursuing the history of the idea of infinity and its use in mathematics, I can recommend “Everything and More” by David Foster Wallace. Miramon has got the relevant information down, though.

Yes, this is a philosophical discussion, but i would point out that nature doesn’t use the digits at all, nor does it count, etc. All of the discussion of measurements and such come from us.

That said, I suspect the issue is our ability to see the very tiny difference between say, pi and 3.14159265.

We can talk about something being arbitrarily big…as in a decimal that doesn’t end, and for any given application, you may need to calculate it further and further out. There are such things in nature. If you are worried about the existence of a perfect circle (though I think we could get one from a soap bubble or a drop of water in a zero g environment, but then I suppose you could worry about the arrangement of the atoms on the surface, or the event horizon of a black hole…) then consider the fibonacci sequence and the golden ratio. You can use the fibonacci sequence to hone in on the golden ratio, and they both occur abundantly in nature. No, they don’t appear to be in the “infinite” form, but they can appear arbitrarily big, and the infinite pattern is made clear…so I think nature is telling us something there, if you will, and we can then infer the rules that give us the irrational number.

There are various schools of philosophy of math, and there are some (or one? it’s been a long time since I’ve discussed philosophy of math at all) that had issues with the infinite, I will admit I had little patience with them, and did not subscribe to their concerns. My point was we are finite creatures, and couldn’t “see” an infinite thing if it smacked us on the head….so we will only perceive a finite piece of it, but that doesn’t mean more isn’t there. (Back to defining the decimal for golden ratio arbitrarily far out…)

KellyM, agreed. We have various situations where our concepts of infinite things can be useful. But they may be concepts that we impose on nature, that are not in fact in nature.

There could be infinities in nature that we would not see if they smacked us on the head, and also the infinities we conceptualize may not be there.

But they are sometimes useful to us, particularly when we try very hard to jam round pegs into square holes and vice versa. Concepts of infinity and infinitesimals may be precisely the bigger hammer we need to smash those other concepts together.

I agree that neither Dühring nor Engels had the mathematical context available to them at the time to really reason well about infinity. That being said, with what was available to them at the time, Dühring continues to be far less than impressive. As Steve said, the crushing of Dühring continues apace.

TO INFINITY AND BEYOND.

Engels’ sarcasm towards Dühring becomes ever clearer: “The need for deliverance is therefore urgent, and by a stroke of good luck Herr Dühring is at hand.”

In the early parts of the chapter, Engels quotes Dühring about the “impermissible contradiction of of a counted infinite numerical series” and proving a

final cause.These things seemed, to me, stupid from the outset, and Engels goes on to be righteously furious with the self-aggrandizing bullshit.There’s a lengthy bit of Dühring that is “copied word for word from a well-known book which first appeared in 1781 and is called

Kritik der reinen Vernunftby Immanuel Kant.” [presumably without attribution by D.] The problem is that Dühring “indefatigably copies down as much of Kant’s antimony as suits his purpose, and throws the rest aside.” So the guy is completely abusing Kant by cherry-picking his plagiarism to make weak, self-defeating arguments which are completely refuted by Kant.Continuing, all of Dühring’s revelations about infinity are based on “axiomatic-tautological grandiloquence.” For instance: “We put forward as a premise precisely what we are to prove.” Later: “Once again, therefore, he puts into the argument, as a premise, the thing that he has to prove.”

This summarizes the really key absurdity: “But all these contradictions and impossibilities are only mere child’s play compared with the confusion into which Herr Dühring falls with his self-equal initial state of the world. If the world had ever been in a state in which no change whatever was taking place, how could it pass from this state to alteration?”

The corner Dühring painted himself into is his belief that the only way to resolve the question of infinity is to say that time had a beginning, which must mean there was an unchanging state of being before time, but he can’t explain how time then began. He tries to use the “bridge of continuity,” which totally

isn’t a thing.He tries on some argument that relies on our present scientific understanding staying static, which is not only inane, it is execrable if you’re scheming to understand the whole world for all time.This chapter held a lot more for me than just the savaging of Dühring (which was, itself, pleasantly witty). Engels used the assorted incorrect methods and conclusions presented by Dühring to illustrate why it is dangerous to a thoughtful worldview to lazily consider ourselves “done” at any point. It is first absurd to claim to know everything, but extraordinarily more so if that supposed knowledge relies on weak theory, hand-waving, or faith.

“The corner Dühring painted himself into is his belief that the only way to resolve the question of infinity is to say that time had a beginning, which must mean there was an unchanging state of being before time, but he can’t explain how time then began”

Okay, well, that’s as clear as I’ve ever heard that expressed.

“Bridge of continuity? Oh, come on. That isn’t even a thing.” You’ve just made me very happy.

“Engels used the assorted incorrect methods and conclusions presented by Dühring to illustrate why it is dangerous to a thoughtful worldview to lazily consider ourselves “done” at any point.”

Ah. True. Good point.

“Okay, well, that’s as clear as I’ve ever heard that expressed.” Excellent. Now I shall be smug about this. :)

As far as we know, the Universe we live in may well be infinite. It’s still being debated, but the existence of actual infinity is not unlikely.

Jonas, agreed. The universe may well be infinite. It’s just hard to get solid evidence either way.

We can conceptually split finite things into an infinite number of pieces, for our own convenience. I argue that when we conceptually split them up that is not an infinity in the world. There could be an infinity in the world that these ideas correspond to, but I haven’t seen the evidence.

We imagine things in the world that are infinitely large, but so far we have not measured any. Our concepts may fit the reality or they may not. Someday we may get evidence that some of the things we thought were infinite were in fact not infinite. I doubt we will get evidence that they are all finite. I doubt we will get evidence that any of them are infinite.

So people debate, because in the absence of evidence it’s either debate the issue or accept that we do not know. And that last is kind of a last resort….