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I have always understood infinite regress to mean going backwards forever. (Forever as in endlessly, not necessarily temporally). A model would be the negative integers, if we viewed them as a model of causation. -1 is "caused" by -2, -2 is caused by -3, -3 is caused by -4, and so forth. In this model, each event has an immediate cause; yet there is no first cause.

For example this is the interpretation of infinite regress in William Lane Craig's Kalam cosmological argument, as I understand it.

Now the SEP article on infinite regress has this exactly backwards:

"An infinite regress is a series of appropriately related elements with a first member but no last member, where each element leads to or generates the next in some sense."

In other words, this looks like induction. A base case and endless succession, like the Peano axioms. The article makes this explicit:

"Peano’s axioms for arithmetic, e.g., yield an infinite regress. "

As I understand it, this is entirely backwards. The article is confusing induction, which has a base case, with infinite regression, which is essentially a recursion or induction without a base case.

I am very confused. My sense is that SEP is simply entirely wrong on this matter, and that my longtime understanding is correct. It's the negative integers that represent infinite regress; and NOT the positive integers. That is: If there is a base case, it's NOT an infinite regress. It's the absence of a base case that defines infinite regress. That's what "regress" means: To go backward.

Set-theoretically, I have always thought that infinite regress is the opposite of well-foundedness. The axiom of foundation is what prevents an infinite regress of set membership. But forward chains of membership x1 ∈ x2 ∈ x3 ∈ ... are perfectly legal, as exemplified by the finite von Neumann ordinals. I know of no one who would call that an "infinite regress." SEP is just wrong.

Any insight? Is the SEP article representing a point of view that's prevalent, and in fact directly opposed to the traditional view of infnite regress as going backward without a starting point? How is it that SEP appears to have this matter entirely wrong? Or is my own understanding wrong all these years?

(Edit) -- Perhaps I didn't make my meaning clear.

My understanding is that infinite regress is a linear order, like this:

... < a4 < a3 < a2 < a1 < a0

It has a last member but no first member. An "infinite regress of causes" is said to be impossible by the Kalam cosmological argument, therefore there must be a first cause, which must be God, etc. The earth sits on a turtle which sits on another turtle which sits on another turtle, without end. It's turtles all the way down.

Contrast that with the structure given by the Peano axioms:

a0 < a1 < a2 < a3 < ...

which has a first element but no last element. There's a big honkin' turtle at the bottom, with another turtle on its back and so forth. It's turtles all the way UP.

Now SEP defines infinite regress as the second case. In my opinion this is 100% wrong. Infinite regress is the first case. The fact that there is an order anti-isomorphism between these two structures, so that the distinction amounts to a renaming and order flipping, is irrelevant. The semantics are completely different. In the SEP model there is a turtle at the bottom. In an infinite regress, it's turtles all the way down.

And of course by induction I did mean mathematical induction, not Humean induction.

Hope this clarifies the intent of my question. Which is: Isn't the SEP article wrong about this?

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  • 2
    Direction doesn't matter. You describe infinite regress as starting from the current element and going down or back without end; SEP describes it as starting from the current element and going up or forward without end. Direction is just a convention, so the two concepts are interchangeable just by calling "next", "previous" or calling "previous", "next". Jul 24 at 22:30
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    @DavidGudeman I see the order-isomorphism but not the semantic meaning. No way that the Peano axioms are an infinite regress. Just as in set theory, forward chains of membership are legal but backward chains are illegal. I thought that was the entire point of infinite regress.
    – user4894
    Jul 24 at 22:32
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    @DavidGudeman The SEP article used vicious without defining it, and I don't think I know what it means. Does it mean unfounded, as in no smallest element? Or circular? Meaning not clear to me.
    – user4894
    Jul 24 at 23:35
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    @DavidGudeman I read the beginning of the Wiki article on vicious infinite regress and didn't understand it at all. I suppose you are right that it doesn't matter whether I start at 0 and go forward or start at zero and go backward. But if you think of time as going in the positive direction, the cosmological arguments are about negative-directed infinite regress. Here's the article, what should I be looking at? en.wikipedia.org/wiki/Infinite_regress#Viciousness
    – user4894
    Jul 24 at 23:42
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    "Vicious" just means that an infinite sequence cannot do what the sequence is needed to do; it means that for some reason, the current element of the sequence depends on the chain of elements having a last element. For example, Hume's Problem is basically that any principle that justifies induction must be justified. And how do you justify it? Any principle you used to justify that principle must also be justified, etc. What we need is an ultimate justification for induction, and Hume's Problem shows that one can't exist. Jul 25 at 1:01
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Imho infinite regress always has a negative connotation (of failure, futility, nonsensicality, …). So I think it means :

  • We have some element (a physical object, a belief, a mathematical construction, …) E0 for which we want to acquire a desired feature F (a full causal explanation, an epistemic justification, well-definedness, …)
  • E0 lacks that feature F. But we could somehow claim it from an E-1 if E-1 has F.
  • But as it turns out E-1 lacks feature F in its own right. It has to be claimed from an E-2, for which it has to be claimed from an E-3, …
  • The resulting series of elements E0, E-1, E-2, E-3, … together with the relation for every pair En, En-1 regarding feature F is the infinite regress.

Of course, the numbering of indices (negative or positive integers) or the words we use (“previous” or “next”), which are associated with “forwards” or ”backwards” direction, is purely a matter of taste.

But mathematical induction is not like an infinite regress as described above. It’s the opposite: we transfer feature F from one element to an ever-growing number of elements. Instead of trying to get feature F for one element from an ever-growing number of elements (this explains the negative connotation).

But the SEP defines infinite regress differently:

An infinite regress is a series of appropriately related elements with a first member but no last member, where each element leads to or generates the next in some sense. An infinite regress argument is an argument that makes appeal to an infinite regress. Usually such arguments take the form of objections to a theory, with the fact that the theory implies an infinite regress being taken to be objectionable.

According to this definition mathematical induction would be an infinite regress. It makes the infinite regress argument look like a fallacy (= usually wrong. There is nothing prima facie objectionable to an infinite regress if defined like in the first sentence).

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I think you are entirely right and that Ross Cameron got it terribly wrong. Cameron confuses two different concepts from order theory. Let's call a strict partial order well-founded, if any of its non-empty subsets possesses a minimal element with respect to the order relation. Let's call a strict partial order right unbounded, if any element has a successor in the ordering.

Cameron openly equates both order-theoretic notions when he expresses the view that the right unboundedness of the natural numbers is "structurally analogous" to the non-well-foundedness of a temporal regression of events. "Structurally analogous" must at least mean that we can find an isomporphism between both orderings. But that of course is terribly wrong, since the natural numbers are well-founded and so have a minimal element, while the event regression is, by its very definition, left unbounded and so has no first element.

I wonder how this SEP-article has made its way into publication, since the order-theoretic confusion mentioned is a sort of beginner's mistake in algebra.

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  • Thanks, that's how I see it. It also turns out that if you click on footnote [1] in the first sentence of the article he says, "Talk of ‘first’ and ‘last’ members here is just a matter of convention" and goes on to point out that it makes no difference. I'm with you, that it makes a profound difference. It's all about well-foundedness. In set theory there are forward chains of set membership; but the axiom of foundation explicitly precludes backward chains.
    – user4894
    Jul 26 at 0:08
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    Concerning set theory (I guess you have something like ZF in mind) the question of course arises if the foundation axiom is really necessary. I'm not quite sure about that. On the one hand, the usual applications of set theory to mathematics do not require the axiom. On the other hand, it guarantees that every set has an ordinal rank and so allows Scott's trick, which gives us sets representing cardinals as equivalence classes.
    – sequitur
    Jul 26 at 23:21
  • Non well-founded sets are studied. It's not that they're impossible; just that set theorists have traditionally made a sharp distinction between forward induction and backward regress, a distinction that the SEP article fails to make. As you point out.
    – user4894
    Jul 26 at 23:26
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The phrase 'infinite regress' means that something is endlessly recursive, not something going endlessly backwards. The classic philosophy-class example is the claim that the earth we stand on is actually the shell of an immense turtle. Then we ask what that this immense turtle is standing on, and learn that it is standing on four other turtles. And of course each of those turtles is standing on four more, and those sixteen third-tier turtles are each standing on four more... It's an endlessly recurring chain of new turtles needed to support the first turtle we presupposed, because that first claim wasn't 'closed': it begged the question of 'what we stand on' entirely, by shifting the ground from our feet to the turtle's feet.

So no, the SEP article is not wrong. The writing isn't the clearest, perhaps, but that's a separate problem. I might have written the first line like so:

An infinite regress is a recursive series where each element requires the existence of the next element for logical completeness, with the result that no element can be invoked without the invocation of an infinitely long string of elements to support it.

Not sure if that's clearer or not; writing philosophy is a PITA...

Induction, by contrast, is a way of avoiding an infinite regress. Technically speaking, if we wanted to know that a theory is true we'd have to test every single one of the infinite number of cases in which the theory would apply, because if even one fails the theory is not 'true' in the absolute sense. Induction says: "I've seen enough successful test cases to assert the theory as true for pragmatic purposes." Rather than pedantically testing case after case after case in the belief that we must ground the last n successful test cases in the results of future test case n+1 (an infinite regress), we instead say we have seen enough and that we will deal with any future failure cases as anomalies to be resolved.

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  • I suspect OP means mathematical induction (a valid proof technique), not empirical induction. And the SEP description “a first member but no last member, where each element leads to … the next” sounds like mathematical induction.
    – viuser
    Jul 25 at 18:15
  • @viuser: The difference between mathematical and empirical induction is minimal; they share the same forward declaration that if x(n) holds, x(n+1) will hold. An infinite regress is the opposite, a reflexive declaration that x(n) holds if and only if x(n+1) holds. Jul 25 at 19:12
  • @viuser Thanks, you're the only one so far who even understands what I'm saying!
    – user4894
    Jul 25 at 19:18
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    @TedWrigley "without the invocation of an infinitely long string of elements to support it." -- But then you are saying 2 supports 1, 3 supports 2, etc. In your model, it's "turtles all the way UP!" Isn't that the opposite of an infinite regress? Please see my edit to my question in case that makes my intent more clear. And of course I did mean mathematical induction, which is completely different than Humean induction. Mathematical induction is a deductive principle, confusingly named perhaps since it's very different than philosophical induction.
    – user4894
    Jul 25 at 20:19
  • @user4894: As I said, mathematical induction and Humean induction are cognates; not different things. Induction says (in English): "If I know that this holds, then the next thing I see after this will also hold." The concept of 'the next thing' is far more rigorously defined in math than in philosophy, obviously, but the intent is the same. Jul 25 at 21:59
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Mathematically, an infinite series may be in one of four classes; having only a first term, only a last term, neither, or both.

The issue at stake here is which of the first three classes we care to describe as "regressive". The English language is not rigorously systematic and words do change their meaning over time, so that modern usage often differs radically from the original meaning of a term. Nowadays, it is commonplace for many kinds of series in all three classes to be referred to as infinite regresses or regressions.

To insist that the term applies to only one of those classes, and assert that SEP (along with a vast array of fellow English-speakers) has "got it badly wrong", is to take no account of its use in modern language.

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  • Cameron equates the structure of the usual ordering of the natural numbers with a non-wellfounded ordering. If that is not badly wrong I don't know what is.
    – sequitur
    Jul 29 at 23:14
  • @sequitur Just because one of many writers makes a slip, does not mean that everybody is clueless. More of a non-sequitur, if you will pardon the allusion to your monicker. Jul 30 at 7:16

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