IH 211 - Infinity

Introduction

What is the infinite? Can we talk about it and understand it?Does such talk produce knowledge?Historically, how have we discussed it?Is the infinite a real existing thing, either material or immaterial? Is it a single concept or many?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

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

What are concepts?Ideas, symbols, meanings, terms, representations [of the world, of other concepts], etc Surely we should define the apparently finite ideas, symbols, etc. before proceeding to the infinite.We form many kinds of concepts. Nobody is certain how we do this, but we do it every day.What about concepts of the infinite?Is infinity only a concept? Humanity wont take no for an answer.Philosophy 101

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

Concepts defined by negative definitions vs. concepts defined by positive definitions.Group A The Negative ConceptsBoundlessness, Endlessness, Unlimitedness, Immeasurability, Eternity, etc.Group B The Positive ConceptsPerfection, Completeness, Wholeness, Absolute Unity, Absoluteness, Self-Sufficiency, Autonomy, etc.Two Concepts of the Infinite

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.

The infinite has influenced our culture; its impact on religion, science and philosophy is especially pronounced.

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.

The methodological approach we will take to analyze the infinite will be chronological. Two Concepts of the Infinite

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 NumbersTwo Concepts of the Infinite

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)

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

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

Dont 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

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

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

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

Example 2:

(1) All Human beings are mammals.(2) Socrates is a Human being.Therefore,(3) Socrates loves Cheese.

This syllogism is formally invalid. Its conclusion is thus not believable.

But how does logic determine what we think of the infinite?Logic

Logic & The InfiniteA strictly logical study of the infinite runs into problems.This is because of the complex nature of the infinite as an idea. If we were to judge an existing item as actually infinite, by what metric would we judge?We really dont know if the hypothesis The infinite actually exists. is sound or unsound. Empiricism upon infinite things unguided by knowledge will yield dubious results.

Zeno (flourished circa 450 B.C.E) was a student of Parmenides, who championed an unchanging universe. Zeno tried to vindicate his teachers views by presenting paradoxes showing the untenable logical consequences that belief in motion and change engender.

Zenos thought experiments were of two kinds: (1) paradoxes ofmotionand (2) paradoxes ofplurality.

Zeno gives the first clearly presented definite idea of an infinite series ().Case Study 1: Zenos Arrow

Zenos 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. We can never say when now really happens.The now as grasped in the present moment instantaneously slips into the not now the very instant you affirm it.The arrow is supposed to be in place A at time A,Since time A is automatically time B then there is no true movement of the arrowCase Study 1: Zenos Arrow

Zenos paradox of motion called The Arrow is where he attempts to prove that an arrow in flight is in reality stationary.

The arrow never really moves it only appears to move.Conclusion: The flying arrow is actually stationary.

Proof: Infinite instants of time can be said to accompany each moment of the arrows flightCase Study 1: Zenos Arrow

How can this conclusion be sound?!

Doesnt 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: Zenos Arrow

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: Zenos Arrow

Forever is composed of nows.

Should Emily Dickinsons assessment of eternity stand?

It appears to require that motion not exist. Can it be modified to avoid Zenos paradox?Discussion Topic

Emily Dickinson, Poet(1830-1886)

Bertrand Russells Response According to the famous 20th century English philosopher Bertrand Russell, Zenos 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

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?Russells Response to Zenos Arrow

The Franciscan Friar Duns Scotus (1266-1388) is known as the Subtle Doctor. He was one of medieval Europes 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: Scotuss Circles

According to Euclidean geometrypoints 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

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.

Scotuss 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

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.Galileos Reply to Scotuss Circles

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.

Galileos paradox is important, since it anticipates later mathematical discoveries concerning mathematically infinite quantities by Georg Cantor. Furthermore, in applying the same logic to Duns Scotuss 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.Galileos Reply to Scotuss Circles

Points, lines, and areasseem to be very differentthings, yet lines andareas are said to becomposed of an infinitenumber of points. Dolines and areas containanything more thanpoints to give them theirdistinctive nature? Cana line or area be madewith a finite number ofpoints. . . for example, acircle?Discussion Topic