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Course 14 · Lesson 3.2

Infinite regress: what is the problem?

10 minute read

Lessons in this course

Course 14 outline

Module 1: Arguments, Existence and Necessity

  1. What natural theology attempts and what it does not
  2. Necessary and contingent existence
  3. How to read and test an argument for God
  4. Philosophy of religion today: a map of the field

Module 2: The Ontological Argument

  1. Anselm and the argument of the Proslogion
  2. Gaunilo and the perfect island objection
  3. Kant: existence is not a real predicate
  4. What the ontological argument would show if it worked

Module 3: Cosmological Arguments

  1. Aquinas: the first three ways
  2. Infinite regress: what is the problem?
  3. Leibniz, the principle of sufficient reason and the argument from contingency
  4. Objections: the brute fact reply and the fallacy of composition

Module 4: The Kalam Cosmological Argument

  1. The kalam tradition: Philoponus, al-Ghazali and Bonaventure
  2. The modern argument: whatever begins to exist has a cause
  3. Did the universe begin? Philosophy and cosmology
  4. Objections and replies, and what follows about the cause

Module 5: Teleological and Fine-Tuning Arguments

  1. From Aquinas’s fifth way to Paley’s watch
  2. Hume and the Dialogues Concerning Natural Religion
  3. Darwin and the change in the design argument
  4. The fine-tuning argument: what is claimed and what is debated
  5. The multiverse and other replies

Module 6: The Moral Argument and Divine Hiddenness

  1. The moral argument: Kant, Lewis and contemporary forms
  2. The Euthyphro question and moral realism without God
  3. Evolution and moral knowledge
  4. The argument from divine hiddenness and Christian replies

Module 7: Contemporary Modal Arguments and Evaluation

  1. Possible worlds and modal logic in plain language
  2. Plantinga’s modal ontological argument
  3. Assessing the modal argument: what it needs and what it shows
  4. Weighing the arguments together, and what none of them prove

All courses

Program / Course 14: Arguments for the Existence of God / Lesson 3.2

What you will learn

  • Define an infinite regress and distinguish harmless from harmful ones.
  • Explain why defenders of cosmological arguments think a regress of dependent causes is a problem.
  • State the objection that some infinite series are coherent.
  • Use the distinction between a series in time and a series of present dependence.

The big idea

A regress is a series in which each member is explained by the one before it, and the series never stops. Some regresses are harmless: the numbers go on forever, and so may the past. Cosmological arguments claim that a regress of things that each get their existence or power from another is different, since nothing in the series ever has anything to give. Critics reply that the whole series may still be enough, and that the argument assumes what it needs.

Key terms

Infinite regress
A series of explanations or causes in which each item depends on a prior one, without a first item.
Vicious regress
A regress that fails to explain what it was brought in to explain, because every step passes the problem back.
Benign regress
A regress that is harmless because no step depends on completing the series.
Actual infinite
A completed collection with infinitely many members, as opposed to a quantity that can always be increased.
Potential infinite
A process that could continue without end, but never reaches an infinite total.
Borrowed power
A way of describing instrumental causes, which pass on a power they receive from another, as a pipe passes on water.
Hilbert’s Hotel
A thought experiment about a hotel with infinitely many rooms, used to show surprising features of the actual infinite.

Background

The question whether an infinite series of causes is possible goes back to Aristotle, who argued that there must be a first mover, and was taken up by Muslim, Jewish and Christian thinkers. It matters because many arguments for God depend on ruling out an endless series of causes or explanations.

Careful writers separate two questions. One is whether the past could be infinite, the subject of the kalam argument in Module 4. The other is whether a series of causes at one time, each depending on the next for its power, could be infinite. Aquinas rejected the second but thought the first could not be settled by reason. The two questions are often confused.

Step-by-step explanation

Step 1: Some infinite series are harmless

The whole numbers go on forever, and nothing is wrong with that. A line can be divided without end. Even a past without a beginning, if there were one, would involve an infinite series of earlier days, each explained by the one before. Many philosophers see no contradiction in this. So the claim that every regress is bad is false, and defenders of cosmological arguments do not make it.

Step 2: A regress that seems to fail

Imagine a chain of train cars, each moving only because the car in front of it pulls it. If there is no engine, what moves the whole train? Adding more cars does not help, because each car is only passing on a pull it receives. Aquinas’s point about the stick, the hand and the stone is of this kind. If every member merely transmits a power it was given, an endless series still has no source.

This is the heart of the claim. The problem is not that the series is long. It is that there is nothing in it that has the power in its own right.

Step 3: The objection from the whole series

Critics reply that the picture of the train begs the question. They say an infinite train might be self-propelling as a whole, or that it is a brute fact that cars are pulled by earlier cars with no further explanation. Why must the power come from a car outside the series? A defender replies that the question what explains the pulling is a fair one and the answer cannot be the same pulling again.

The disagreement turns on whether every series must have an explanation, which leads toward the principle of sufficient reason in the next lesson.

Step 4: The paradoxes of the actual infinite

Some defenders argue that an actual infinite cannot exist in the real world. They cite Hilbert’s Hotel, in which a full hotel with infinitely many rooms can still take new guests by moving each guest to the next room. This seems absurd to them. Critics reply that the paradoxes show only that infinity behaves differently from finite numbers, as mathematicians have long known, and that mathematics uses actual infinities without contradiction. The question whether real things can be infinite is disputed.

Step 5: What the regress argument can show

If the regress objection works, it shows only that a first member exists, not that this first member is God. The step from first cause to God needs further argument, which comes from the attributes that a being without dependence would have: it would be unchanging, unlimited and the source of all else. Aquinas supplies these in later questions.

  1. What difference is there between a long series and a series with nothing in it that has power in its own right?
  2. Can you think of a regress that seems harmless to you?
  3. If every cause is itself caused, does the series need a further explanation?
  4. Do you find Hilbert’s Hotel absurd, or only surprising?
  5. Why might the past be infinite while the dependence of things now cannot be?
  6. What would count as an explanation of an infinite series?

Examples

The borrowed book

A asks B for a book, B asks C, C asks D, and so on without end. No one has the book. However long the chain, the book never arrives. Defenders say causal regresses of borrowed power are like this. (A constructed example.)

A series of days

If time had no beginning, each day would be preceded by another. Each day does not need the one before it to continue to exist now. This is a harmless regress, and Aquinas thought it could not be ruled out by reason alone.

Hilbert’s Hotel

A hotel has infinitely many rooms, all full. A new guest arrives, and the manager asks each guest to move to the next room, leaving room 1 free. The paradox is meant to show that actual infinities behave strangely. Mathematicians accept the result as a feature of infinite sets.

Different perspectives

Catholic

Aquinas taught that an essentially ordered series cannot be infinite, but that an infinite past is not shown impossible by reason. This remains the standard Catholic philosophical teaching, though the Church has taught by faith that the world had a beginning in time (Fourth Lateran Council, 1215).

Orthodox

Orthodox tradition teaches that the world is created out of nothing and depends continually on God. It does not usually argue from regress. John of Damascus used cosmological arguments in a brief form, but the typical Orthodox stress is on the world’s constant dependence on God’s energies.

Protestant

Protestants vary. Craig, an evangelical, argues that an actual infinite cannot exist and uses this in the kalam argument. Reformed thinkers like Plantinga have not stressed the regress. Some Protestant scholars follow Aquinas in distinguishing the two kinds of series.

Scholarly views

Contemporary philosophers disagree. Some, such as Alexander Pruss, defend versions of the argument for a first member. Others, such as Oppy, argue that infinite series are coherent and that no first member is needed. Mathematicians and philosophers of mathematics generally accept actual infinities in mathematics.

Skeptical views

Skeptics say that the intuition that something must be at the start of a series is not reliable beyond ordinary experience. They add that physics already shows features of time and space that common sense does not expect.

Why it matters

Almost every cosmological argument stands or falls on the treatment of regress. A student who can state exactly which kind of regress is claimed to be vicious, and why, will be able to read those arguments without confusion.

Common misunderstandings

Misunderstanding: Believers think that any infinite series is impossible.

Correction: Aquinas and many others accept that some infinite series, such as numbers, are harmless. They deny it of series in which each member depends on another for its power now.

Misunderstanding: Hilbert’s Hotel proves that the past cannot be infinite.

Correction: It is used to argue this, but mathematicians point out that it shows only that infinite collections behave unlike finite ones. Its force is disputed.

Misunderstanding: If there is a first cause, it must be God.

Correction: A first cause shows something about the structure of dependence. It does not alone show that it is personal, good or worthy of worship.

Misunderstanding: Critics think the world came from nothing.

Correction: Most critics hold either that the series is infinite or that the universe is a brute fact. Neither claim is the same as coming from nothing.

Knowledge check

What is a vicious regress?

It is a series that fails to explain what it was brought in to explain, because each step passes the question to the next. In a chain of borrowed power, an endless series of members that each need a push has no source of the push. A harmless regress, like the whole numbers, does not depend on completing the series to explain anything.

Why did Aquinas distinguish two kinds of series?

In one kind, each member acts only because the member above it is acting now, so without a first member nothing acts. In the other, each member produced the next and no longer needs it. Aquinas held that only the first kind cannot be infinite, and he thought an endless past could not be ruled out by reason.

What is the main objection to the regress argument?

The objector says the argument assumes that the whole series needs an explanation outside itself. An endless train might simply be a brute fact, or the whole might be self-sustaining. Defenders respond that merely repeating the same pulling does not explain the pulling. The disagreement turns on whether every series needs an explanation.

What does Hilbert’s Hotel try to show, and how is it answered?

It tries to show that an actual infinite leads to absurd results, such as a full hotel that can still take new guests. Critics answer that this is a known feature of infinite sets in mathematics and involves no contradiction. They say that strange is not the same as impossible, so the example does not settle the question.

Think it through

  1. Do you think the whole of reality needs an explanation, or can some things simply be?
  2. Is it easier to accept an endless chain or a first member that needs no cause?
  3. Does the example of borrowed power match anything in physics as you know it?
  4. Why do you think people feel strongly about this question?

Independent assignment

Harmless and harmful regresses

In your own notebook, write 300 to 450 words. List three regresses that seem harmless and three that seem harmful, with one sentence for each. Then write a paragraph stating the difference you see between the two groups. Finish with the strongest objection to your own account. Do not send this to anyone.

Further study