Mathematics and the Infinite (part 2)

Regina Cœli Academy
Natural Philosophy – Physics
Lecturer: Mr. Alan Aversa

O Creator ineffable, who of the riches of Thy wisdom didst
appoint three hierarchies of Angels and didst set them in
wondrous order over the highest heavens, and who didst
apportion the elements of the world most wisely: do Thou,
who art in truth the fountain of light and wisdom, deign to
shed upon the darkness of my understanding the rays of
Thine infinite brightness, and remove far from me the twofold
darkness in which I was born, namely, sin and ignorance. Do
Thou, who givest speech to the tongues of little children,
instruct my tongue and pour into my lips the grace of Thy
benediction. Give me keenness of apprehension, capacity for
remembering, method and ease in learning, insight in
interpretation, and copious eloquence in speech. Instruct my
beginning, direct my progress, and set Thy seal upon the
finished work, Thou, who art true God and true Man, who
livest and reignest world without end. Amen.
(St. Thomas Aquinas Oratio ante studium)

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Russell Reduces Infinity into a
Logical Hierarchy
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Russell Reduces Infinity into a
Logical Hierarchy

We will discuss three philosophies of
mathematics:
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Russell believed that his logicalism would have to be
supplemented by empirical data.

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Kurt Gödel: “the foundations of mathematics must always
contain at least a small number of undefined terms [‘the
undecidables’] which cannot be treated in the system
which they organize.”

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Gödel thus proved both logicalism and formalism false ∵
these undecidables will preclude them “from ever saying
the final word about reality or mathematics.”

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Hilbert responded to Gödel by saying that
mathematics need not conform to reality.
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“Content is sacrificed to form, inner meaning to mere rule.”

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Formalism Makes Infinity into a Rule

Hilbert dismisses the need for the empirical.
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Thus, e.g., ‫ ∅א‬is finite “in the structure of its definition,” although it
represents the infinite series of algebraic numbers.

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Hilbert’s formalism takes logicalism to the extreme.
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In this example, “logical types” put the Cretan outside.

Infinity need not be infinitely complex.

Brouwer, Weyl, and Poincaré
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But technically infinity is not a number.

Russell resolved this with “logical types.”
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David Hilbert and his school

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This is Russell’s “paradox of the infinite.”

Example paradox: “A Cretan says: ‘All Cretans are liars.’”
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Formalism Makes Infinity into a Rule
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If number is the “class of similar classes,” then infinity would
have to include itself as a member!
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We briefly discussed this last time.
– Number is “the class of similar classes.”

Intuitionism
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Russell’s goal is “to make the modern method more
discursive and, like matter, more controllable.”

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Formalism
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of Peano and Frege, culminating in Russell
Reduction of mathematics to logic
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Logicalism
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Mathematics for Hilbert is just a manipulation of
symbols.

Hilbert: “A number is not an object in the proper
sense but a property.”
Infinity is an idea.
Inductive side of mathematics: “One formula [of
mathematics] is conceived and derived from
another.”

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Intuitionism Denies Hierarchy of
Infinites

Formalism Makes Infinity into a Rule
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“When we generalize this situation, then mathematics arrives
at a number of formulas, first those which correspond to the
matter expressed in finite propositions, in essentially numbered
equalities and inequalities, and second at broader formulæ that
in themselves have no meaning and form the ideal
representations of our theory.” (Hilbert’s Grundlagen der
Geometrie)
Infinity is in thought alone.
Hilbert: “The infinite divisibility of the continuum is an operation
which exists in thought only, is just an idea, an idea which is
refuted by our observations of nature, as well as by physical
and chemical experiments.”

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“Objectively and subjectively, a thing is prior to its
relations. One is what it is, first, and is then related
to two, and the order of thought follows along with
this arrangement of reality itself.”

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Brouwer believes, however, one and two are “drawn
much more tightly together.”

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“[I]n a single flash of insight, there arise the
concepts of one, two, and their relation.”

Intuitionists recognize only denumerable sets.

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“The primary intuition, the bedrock of all
mathematics, is the ‘intuition of the bare twooneness’.” (Brouwer, “Intuitionism and
Formalism,” Bul. AMS, XX (1913) 85-86)

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“The notion of the infinite is precipitated from
considering this idea of ‘between’ which can be
strewn indefinitely with repetitions of the ‘twooneness.’”
“No infinity can be finited to a point of being
compared with other infinities” because Brouwer
considers this opposed to his “two-oneness” intuition.
Thus, Brouwer only admits the ‫ ∅א‬infinity.

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Intuitionists emphasize mathematical content;
formalists “emphasize the regulative and
logical.”

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Kronecker’s finitism: “God made the integers and man
everything else.”
“the notion of all the real numbers between 1 and 2”:
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Intuitionists reject Cantorian infinities ∵ they
have “content” only “among the rules of logic.”

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Intuitionists reject the “domination by the logical
order.”
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Intuitionism Denies Hierarchy of
Infinites

Mathematics has a priority over logic in
intuitionism.

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Brouwer can indefinitely “split” the “two-oneness” to
form the concept of “between.”

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Intuitionism Denies Hierarchy of
Infinites
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Mathematics must begin with the intuitive.

Intuitionism Denies Hierarchy of
Infinites

Brouwer: “intuitions of the bare two-oneness”
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Kant thought Euclidean geometry was a priori

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Infinites
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Intuitionism results from Kant’s synthetic a priori
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“Infinity here exists only logically.”

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Intuitionists: L. E. J. Brouwer, Herman Weyl,
Leopold Kronecker, Henri Poincaré

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Russel et al. formalists: the “middle area is mapped out in
terms of a series of infinite decimals, requiring an infinite
number of operations.”
Intuitionists: “the stretch from 1 to 2 is spanned by a law to
construct the series of intervening decimals, a law that is
involved by the analysis of ‘between’; there is thus a finite
number of operations.”
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“no such thing as a more ‘between’ or a less ‘between.’”
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Intuitionism Denies Hierarchy of
Infinites
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Intuitionism denies the principle of the excluded
middle.
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Intuitionism Denies Hierarchy of
Infinites

“A thing is either A or not-A, without the possibility
of a middle ground.”

It holds instead a “three-valued logic” or “threevalued mathematics”:
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References
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V. E. Smith’s Philosophical Physics
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Please continue reading ch. 9 (Mathematics and
the Infinite).
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The required reading will be in the reginacoeli.box.com
account.

Have a blessed summer break!

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1) Concrete individual judgment (e.g., 2 + 5 = 7), from
which intuitions are obtained
2) Symbolic stage (e.g., m + n = n + m)
4) From this, any number can be formed ad infinitum.

Is this indeterminate Aristotle’s potency, the intermediary
between being and non-being?

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Hermann Weyl succinctly outlines the
intuitionists’ understanding of the infinite in four
steps:

3) Concept of sum in general obtained

“either A or not-A or indeterminate”
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