A philosophical introduction to the problem of understanding the infinite.

1. Introduction

2. o What is the infinite?
o Can we talk about it and understand it?
o Does such talk produce knowledge?
o Historically, how have we discussed it?
o Is the infinite a real existing thing, either material or
immaterial?
o Is it a single concept or many?
o If the infinite is a real and existing thing then can we
grasp and know infinity merely by discussing or
thinking about it?
Inquiry Into The Infinite Creates
More Questions Than Answers

3. o “The infinite is that for
which it is always possible
to take something outside.”
o This is one philosophical
concept defining the infinite
in terms of other things,
namely, what we already
know or have.
Aristotle’s Proposed Infinity

4. o What are concepts?
o Ideas, symbols, meanings, terms, representations [of
the world, of other concepts], etc…
o Surely we should define the apparently finite
ideas, symbols, etc. before proceeding to the
infinite.
o We form many kinds of concepts. Nobody is certain
how we do this, but we do it every day.
o What about concepts of the infinite?
o Is infinity only a concept?
o Humanity won’t take “no” for an answer.
Philosophy 101

5. Philosophy 101
o A concept, for us, is connected with thought.
o Thought is about the world, but the entities in the
world are not thoughts or ideas.
o What is the connection between thoughts & things?
o The answer depends on the framework one uses.
o For the conceptualist, ideas or thoughts are all we
can know. Concepts are reality.
o For the realist, our concepts or thoughts are
dependent on transcendent entities. The things in
themselves come first.

6. o Concepts defined by negative definitions vs. concepts
defined by positive definitions.
o Group A – The Negative Concepts
o Boundlessness, Endlessness, Unlimitedness,
Immeasurability, Eternity, etc.
o Group B – The Positive Concepts
o Perfection, Completeness, Wholeness, Absolute
Unity, Absoluteness, Self-Sufficiency, Autonomy,
etc.
Two Concepts of “the Infinite”

7. “The concepts in [A] are more negative and
convey a sense of potentiality. They are the
concepts that might be expected to inform a more
mathematical or logical discussion of the infinite.
The concepts in the second cluster [B] are more
positive and convey a sense of actuality. They are
the concepts that might be expected to inform a
more metaphysical or theological discussion of
the infinite”
A. W. Moore, The Infinite (New York: Routledge Press) 1990, p. 2.

8. o The infinite has influenced our culture; its impact on
religion, science and philosophy is especially
pronounced.
o Because the concept of infinity is so broad and
complex the best approach to take is, arguably, the
philosophical one; but with a special emphasis on
historically relevant scientific or quantitative
(physical and mathematical) applications.
o The methodological approach we will take to analyze
the infinite will be chronological.
Two Concepts of “the Infinite”

9. o We will cover 4 broad historical periods.
• Infinity in the Ancient World
• Medieval Conceptions of the Infinite
• Renaissance Conceptions of the Infinite
• The Infinite in Modernity – Scientific
Applications & Transfinite Numbers
Two Concepts of “the Infinite”

10. Philosophy is, amongst other things, the intellectual
discipline that studies argumentation and seeks the truth
about the world and our place in it.
Philosophy is the ideal discipline to use in approaching
the study of basic or essential concepts because only
philosophy can connect different disciplines together in
order to arrive at a broader perspective and clarify our
understanding in a fundamental way.
Philosophy 101 (Part 2)

11. Philosophers rely on two essential tools in order to
formulate theories and arrive at a deeper understanding
of the world:
Philosophy 101 (Part 2)
LOGIC Imagination

12. Logic, in its broadest sense, is reasoning that includes both formal
and informal logic. Informal logic is used everyday. Formal logic
attempts to reduce all claims and judgement of truth to
manipulation of symbols by means of rules.
As a theoretical discipline, logic is normative (seeks what should
or ought to be) and describes ideal forms of thought that, when
valid, are truth preserving.
Applying logic means analyzing the structure of an argument. By
(formally) studying the structure of thought we can explore the
coherence and rational properties of reasoned arguments.
Logic

13. Don’t confuse logic with philosophy as a whole. Logic is a tool, an
instrument that allows us to clarify, apply, organize and increase
knowledge.
It is not an end in itself and cannot determine, without assistance,
how concepts describe the world or whether they reflect reality.
Most importantly, logic is no substitute for judgement, which is a
personal responsibility. Thus, logic cannot judge the ultimate truth
or falsity of concepts.
Logic

14. Since logic has truth as its normative goal, it is an essential part of
science.
But, not all science is strictly formal.
Induction and experimentation make use of assumptions and
methods that are informal but also indispensible to modern
science.
Logic

15. In most logical systems, all arguments are either: valid or invalid
(i.e. they either have some statement that contradicts another
requirement within the network of claims or they avoid
contradiction).
All arguments have premises that are either sound (true and
believable) or unsound (untrue or not believable).
Logic cannot establish soundness;- it must presuppose it.
Therefore, the concepts and fundamental claims that enter into any
logical system will determine whether the fundamental soundness
of its conclusions.
Logic

16. Example 1:
(1) The world (and the totality of things in the physical
universe) is made of cheese.
(2) John is a part of the world.
Therefore,
(3) John is made out of cheese.
While this syllogism is formally valid, it is not believable.
Logic

17. Example 2:
(1) All Human beings are mammals.
(2) Socrates is a Human being.
Therefore,
(3) Socrates loves Cheese.
This syllogism is formally invalid. It’s conclusion is thus not
believable.
But how does logic determine what we think of the infinite?
Logic

18. Logic & The Infinite
o A strictly logical study of the infinite runs into
problems.
o This is because of the complex nature of the infinite
as an idea.
o If we were to judge an existing item as actually
infinite, by what metric would we judge?
o We really don’t know if the hypothesis “The infinite
actually exists.” is sound or unsound.
o Empiricism upon infinite things unguided by
knowledge will yield dubious results.

19. Zeno (flourished circa 450 B.C.E) was a student of Parmenides,
who championed an unchanging universe. Zeno tried to vindicate
his teacher’s views by presenting paradoxes showing the
untenable logical consequences that belief in motion and change
engender.
Zeno’s thought experiments were of two kinds: (1) paradoxes
of motion and (2) paradoxes of plurality.
Zeno gives the first clearly presented definite idea of an infinite
series (άπειρον).
Case Study 1: Zeno’s Arrow

20. Zeno’s paradox of motion called “The Arrow” is where he
attempts to prove that an arrow in flight is in reality stationary.
Hypothesis: Time is composed of instantaneous moments.
1. We can never say when ‘now’ really happens.
2. The ‘now’ as grasped in the present moment instantaneously
slips into the ‘not now’ the very instant you affirm it.
3. The arrow is supposed to be in place A at time A,
4. Since time A is automatically time B then there is no true
movement of the arrow
Case Study 1: Zeno’s Arrow

21. Zeno’s paradox of motion called “The Arrow” is where he
attempts to prove that an arrow in flight is in reality stationary.
6. The arrow never really moves it only appears to move.
7. Conclusion: The flying arrow is actually stationary.
Proof: Infinite instants of time can be said to accompany each
‘moment’ of the arrows flight
Case Study 1: Zeno’s Arrow

22. How can this conclusion be sound?!
Doesn’t It contradicts the evidence of our experience that time
‘flows’?
Time seems to flow at different rates, how do we know that there
is an objective time, i.e. a single time that flows uniformly and is
the same for everyone?
To complicate our reliance on common sense notions of time, if
Albert Einstein and modern physics are correct (covered in later
chapters), absolute time and space must be abandoned, i.e. from a
physical point of view.
Case Study 1: Zeno’s Arrow

23. How can this conclusion be sound?!
Ignoring the deeper problems of our ignorance regarding the
reality of time and space –
Is the argument Zeno gives valid? What about sound?
Case Study 1: Zeno’s Arrow

24. “Forever is composed of nows.”
Should Emily Dickinson’s
assessment of eternity stand?
It appears to require that motion
not exist. Can it be modified to
avoid Zeno’s paradox?
Discussion Topic
Emily Dickinson, Poet
(1830-1886)

25. Bertrand Russell’s
Response
According to the famous 20th
century English philosopher
Bertrand Russell, Zeno’s arrow
argument is an ad hominem
argument (an attack on those
defending motion as real) and can
be summarized as follows:
“[the arrow] is never moving, but in some
miraculous way the change of position has to
occur between the instants, that is to say, not at
any time whatever”

26. How can this conclusion be sound?!
Ignoring the deeper problems of our ignorance regarding
detailed mechanics of time and space –
Is the argument Zeno gives valid?
Russell’s Response to Zeno’s
Arrow

27. The Franciscan Friar Duns Scotus (1266-1388) is known as ’the
Subtle Doctor’. He was one of medieval Europe’s greatest minds.
Like many medieval philosophers, Scotus philosophized against
the backdrop of a strong theistic context. Scotus argued, against
prevailing Scholastic assumptions, that our inability to conceive
of an actual mathematical infinity actually "reflects. . . our own
limits" rather than any inherent limitations pertaining to God or
reality
Case Study 2: Scotus’s Circles

28. According to Euclidean geometry
points have no magnitude and a circle
consists of an infinite series of points.
Scotus, using two circles, shows how both
must be constructed out of an infinite series
of points yet our perception will reveal that
the two circles are not quantitatively infinite
in the same way.
Discussion Topic

29. The outer circle is clearly larger than the
inner. This leads to the result that point A’ on
the large circle corresponds exactly to point
A on the smaller circle and point B’ to B.
Since geometrical points have no extension,
this creates a problem comprehending how
this correspond-ence can exist.
Scotus’s solution is that there are two infinite
sets of points that should be the same size but
are also different in size. This, of course,
leads to a paradox.
Discussion Topic

30. While still a professor of mathematics in Padua, Italy in the
1600s, attempted to respond to Duns Scotus’ paradox of the two
circles. He argues,
“[if] I inquire how many [square] roots there are, it cannot be
denied that there are as many as the numbers because every
number is the root of some square. This being granted, we must say
that there are as many squares as there are numbers because they
are just as numerous as their roots, and all the numbers are roots.
Yet at the outset we said that there are many more numbers than
squares, since the larger portion of them are not squares. Not only
so, but the proportionate number of squares diminishes as we pass
to larger numbers… if one could conceive of such a thing, he would
be forced to admit that there are as many squares as there are
numbers taken all together.”
Galileo’s Reply to Scotus’s Circles

31. Galileo concludes: “So far as I see we can only infer that the
totality of all numbers is infinite, that the number of squares is
infinite, and that the number of their roots is infinite; neither is the
number of squares less than the totality of all the numbers, nor the
latter greater than the former; and finally the attributes "equal,"
"greater," and "less,“ are not applicable to infinite, but only to finite,
quantities.”
Galileo’s paradox is important, since it anticipates later
mathematical discoveries concerning mathematically infinite
quantities by Georg Cantor. Furthermore, in applying the same
logic to Duns Scotus’s circles, Galileo concluded that although the
number of points used to construct the two circles was the same
size (infinite) one of the circles appears larger because of
infinitely small gaps.
Galileo’s Reply to Scotus’s Circles

32. Points, lines, and areas
seem to be very different
things, yet lines and
areas are said to be
composed of an infinite
number of points. Do
lines and areas contain
anything more than
points to give them their
distinctive nature? Can
a line or area be made
with a finite number of
points. . . for example, a
circle?
Discussion Topic