Mathematics - Revision history

David Persyn: branding

2011-07-26T21:02:25Z

branding

David Persyn: Protected "Mathematics" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

2011-07-16T17:29:24Z

Protected "Mathematics" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

David Persyn: /* An Analytical Contradiction in "Infinite Past Time" */ format pseudo-table

2011-07-16T12:58:25Z

‎An Analytical Contradiction in "Infinite Past Time": format pseudo-table

David Persyn: /* Footnotes */ remove temp unlinked footnotes

2011-07-16T12:56:46Z

‎Footnotes: remove temp unlinked footnotes

David Persyn: quotes and footnotes

2011-07-16T12:56:02Z

quotes and footnotes

David Persyn: /* Summary of Hilbert's Prohibition of Actual Infinities */ quotes and footnotes

2011-07-16T12:50:39Z

‎Summary of Hilbert's Prohibition of Actual Infinities: quotes and footnotes

David Persyn: /* A Brief Explanation of Real Time */ quotes and footnotes

2011-07-16T12:44:53Z

‎A Brief Explanation of Real Time: quotes and footnotes

David Persyn: /* An Analytical Contradiction in "Infinite Past Time" */ footnote

2011-07-16T12:39:33Z

‎An Analytical Contradiction in "Infinite Past Time": footnote

David Persyn: remove all caps line

2011-07-16T01:40:43Z

remove all caps line

David Persyn: /* Footnotes */ Raw Footnotes (not yet linked)

2011-07-16T01:05:40Z

‎Footnotes: Raw Footnotes (not yet linked)

← Older revision		Revision as of 21:02, 26 July 2011
Line 1:		Line 1:
−	~~MOERF – GOD UNIT G~~	+	The Magis God Wiki: Mathematics

	Is There Evidence of God From the Philosophy of Mathematics??		Is There Evidence of God From the Philosophy of Mathematics??

@@ Line 17: / Line 17: @@
 Now let us turn to the other side of the expression, namely, “infinite.”  Throughout this Unit, I will view “infinity” within the context of a continuous succession because I will show that real time in changeable universes must be a “continuous succession of non-contemporaneous distension.”  Now, infinities within a continuous succession imply “unoccurrable,” “unachievable,” and “unactualizable,” for a continuous succession occurs one step at a time (that is, one step after another), and can therefore only be increased a finite amount.  No matter how fast and how long the succession occurs, the “one step at a time” or “one step after another” character of the succession necessitates that only a finite amount is occurrable, achievable, or actualizable.  Now, if “infinity” is applied to a continuous succession, and it is to be kept analytically distinct from (indeed, contrary to) “finitude,” then “infinity” must always be more than can ever occur, be achieved, or be actualized through a continuous succession (“one step at a time” succession).  Therefore, infinity would have to be unoccurrable, unachievable, and unactualizable when applied to a continuous succession.  Any other definition would make “infinity” analytically indistinguishable from “finitude” in its application to a continuous succession.  Therefore, in order to maintain the analytical distinction between “finitude” and “infinity” in a continuous succession, “infinity” must be considered unoccurrable (as distinct from finitude which is occurrable), unachievable (as distinct from finitude which is achievable), and unactualizable (as distinct from finitude which is actualizable).  We are now ready to combine the two parts of our expression through our three common conceptual bases:
-“Infinite..............................Past Time”
+'''“Infinite..............................Past Time”'''
-“(The) unoccurable.............(has) occurred.”
+“(The) unoccurable................(has) occurred.”
-“(The) unachievable............(has been) achieved.”
+“(The) unachievable...............(has been) achieved.”
-“(The) unactualizable..........(has been) actualized.”
+“(The) unactualizable.............(has been) actualized.”
 Failures of human imagination may deceive one into thinking that the above analytical contradictions can be overcome, but further scrutiny reveals their inescapability.  For example, it might be easier to detect the unachievability of an infinite series when one views an infinite succession as having a beginning point without an ending point, for if a series has no end, then, a priori, it can never be achieved.  However, when one looks at the infinite series as having an ending point but no beginning point (as with infinite past time reaching the present), one is tempted to think that the presence of the ending point must signify achievement, and, therefore, the infinite series was achieved.  This conjecture does not avoid the contradiction of “infinite past time” being “an achieved unachievable.”  It simply manifests a failure of our imagination.  Since we conjecture that the ending point has been reached, we think that an infinite number of steps has really been traversed, but this does not help, because we are still contending that unachievability has been achieved, and are therefore still asserting an analytical contradiction.

← Older revision		Revision as of 12:56, 16 July 2011
Line 138:		Line 138:

	<references />		<references />
−
−	~~“Real time” will be defined as “a continuous succession of non-contemporaneous distension.”~~
−
−	~~Bergson 1965, p. 48. See also Spitzer 1989, pp. 12-14.~~
−
−	~~Bergson 1965, p. 48.~~
−
−	~~Beginning with Pseudo-Dionysius the Areopagite. See Rolt 1940.~~
−
−	~~Hilbert 1964, p. 135.~~
−
−	~~Hilbert 1964, p. 141.~~
−
−	~~Hilbert 1964, p. 151.~~
−
−	~~Craig 1993, p. 10. Many other mathematicians and philosophers have recognized this as well. See, for example, Whitrow, 1954-1955, pp. 215-225; 1967, pp. 422-432; 1968; and 1970, pp. 224-233.~~
−
−	~~Rotman and Kneebone 1966, p. 61. Italics mine.~~

@@ Line 112: / Line 112: @@
 Hilbert then concludes that the technique of ideal elements (which can imply infinities) cannot be used if they change the fundamental axioms of finite numbers to which they have been applied.  Since this does not occur with potential infinities (B-infinities), but always occurs with actual infinities (C-infinities), Hilbert rejects the use of the latter in any way that could apply to the real world (i.e., real magnitudes, real counting, real series, etc.):
 == Conclusions ==
 Hilbert’s analysis shows that the existence of a mathematical infinity in an algorithmically finite structure results not only in a contradiction, but also in the undermining of the axioms of finite numbers which it was intended to complete.  If devastating consequences for the whole of mathematical reasoning are to be avoided, C-infinities must not be applied to real magnitudes, successions, series, or any algorithmically finite structure that could be considered real (such as past time).
@@ Line 121: / Line 121: @@
 Hilbert’s prohibition of C-infinities continues to be widely held by contemporary mathematicians.  As William Lane Craig notes:
-According to Robinson, “Cantor’s infinities are abstract and divorced from the physical world [Robinson 1969, p. 163].”  This judgement is echoed by Fraenkel, who concludes that among the various branches of mathematics, set theory is “the branch which least of all is connected with external experience and most genuinely originates from free intellectual creation [Fraenkel 1973, p. 240].”  As a creation of the human mind, state Rotman and Kneebone…when one selects from an infinite set an infinite subset, the actual possibility of such an operation is not implied.  “The conception of an infinite sequence of choices (or of any other acts)…is a mathematical fiction – an idealization of what is imaginable only in finite cases [Rotman and Kneebone 1966, p. 60].”
+<blockquote>According to Robinson, “Cantor’s infinities are abstract and divorced from the physical world [Robinson 1969, p. 163].”  This judgement is echoed by Fraenkel, who concludes that among the various branches of mathematics, set theory is “the branch which least of all is connected with external experience and most genuinely originates from free intellectual creation [Fraenkel 1973, p. 240].”  As a creation of the human mind, state Rotman and Kneebone…when one selects from an infinite set an infinite subset, the actual possibility of such an operation is not implied.  “The conception of an infinite sequence of choices (or of any other acts)…is a mathematical fiction – an idealization of what is imaginable only in finite cases [Rotman and Kneebone 1966, p. 60].” <ref>Craig 1993, p. 10. Many other mathematicians and philosophers have recognized this as well. See, for example, Whitrow, 1954-1955, pp. 215-225; 1967, pp. 422-432; 1968; and 1970, pp. 224-233.</ref></blockquote
 Of course, infinities can be applied to sets in merely theoretical or abstract ways (e.g., Cantorian sets or the Zermelo-Fraenkel universe of sets), but this cannot be thought to have applicability to the real world:
-[T]he Zermelo-Fraenkel universe of sets exists only in a realm of abstract thought…  [I]he “universe” of sets to which the…theory refers is in no way intended as an abstract model of an existing Universe, but serves merely as the postulated universe of discourse for a certain kind of abstract inquiry.
+<blockquote>[T]he Zermelo-Fraenkel universe of sets exists only in a realm of abstract thought…  [I]he “universe” of sets to which the…theory refers is in no way intended as an abstract model of an existing Universe, but serves merely as the postulated universe of discourse for a certain kind of abstract inquiry.<ref>Rotman and Kneebone 1966, p. 61.</ref></blockquote>
 In sum, Hilbert’s, Fraenkel’s, Rotman’s, Kneebone’s, Zermelo’s, Robinson’s, and many others’ analysis shows that the existence of a mathematical infinity in an algorithmically finite structure results not only in a contradiction, but also in the undermining of the axioms of finite numbers which it was intended to complete.  If devastating consequences for the whole of mathematical reasoning (and also the applicability of mathematics to the finite universe) are to be avoided, C-infinities must not be applied to real magnitudes, successions, series, or any algorithmically finite structure that could be considered real (such as past time).

@@ Line 77: / Line 77: @@
 The Hilbertian prohibition applies to the C-infinity alone, for, as will be seen, it is not concerned with non-mathematical infinities (i.e., an A-infinity), and it permits indefinitely ongoing (continually potential) successions through algorithmically finite structures (i.e., B-infinities).  Before showing Hilbert’s and others’ prohibition of C-infinities, the two permissible kinds of infinities will be discussed.
-An A-infinity has long been recognized by the Scholastic tradition.  As noted above, it is not a mathematical infinity (such as infinite sets, infinite number lines, infinite successions, etc.) and it is not applied to algorithmically finite structures (such as spatial magnitudes, temporal magnitudes, fields, forces, etc.).  Hence, it does not postulate an infinite Euclidean plane, infinite past time, an infinite number line, infinite space, infinite history, infinite thermometers, infinite density, or an infinite physical force.  An A-infinity is simply the recognition of “non-restrictedness” in power.  It is, therefore, a negation of any predicate which has restriction or could imply restriction in an infinite power.
+An A-infinity has long been recognized by the Scholastic tradition.<ref>Beginning with Pseudo-Dionysius the Areopagite. See Rolt 1940.</ref>  As noted above, it is not a mathematical infinity (such as infinite sets, infinite number lines, infinite successions, etc.) and it is not applied to algorithmically finite structures (such as spatial magnitudes, temporal magnitudes, fields, forces, etc.).  Hence, it does not postulate an infinite Euclidean plane, infinite past time, an infinite number line, infinite space, infinite history, infinite thermometers, infinite density, or an infinite physical force.  An A-infinity is simply the recognition of “non-restrictedness” in power.  It is, therefore, a negation of any predicate which has restriction or could imply restriction in an infinite power.
 As Scholastic philosophers have long recognized,  one can only speak about “infinite power” or “infinite being” by negating any restriction (or structure giving rise to a restriction such as a divisible magnitude) to the power itself.  Thus, one can say that “infinite power” is not restricted as to form, way of acting, space-time point, or even to spatiality itself (which is a divisible magnitude having intrinsically finite parts).
@@ Line 105: / Line 105: @@
 The above discussion was brought to the attention of philosophers of mathematics by David Hilbert, who attempted to clarify the notion of an infinite numeric series which was thought to exist as a completed totality:
-Just as in the limit processes of the infinitesimal calculus, the infinite in the sense of the infinitely large and the infinitely small proved to be merely a figure of speech, so too we must realize that the infinite in the sense of an infinite totality, where we still find it used in deductive methods, is an illusion.
+<blockquote>Just as in the limit processes of the infinitesimal calculus, the infinite in the sense of the infinitely large and the infinitely small proved to be merely a figure of speech, so too we must realize that the infinite in the sense of an infinite totality, where we still find it used in deductive methods, is an ''illusion''.<ref>Hilbert 1964, p. 135.</ref></blockquote>
 Hilbert is proposing here that, even though a B-infinity (one with the potential to continue indefinitely without being actual) is mathematically admissible, a C-infinity (the existence of a mathematical infinity within an algorithmically finite structure) is not mathematically admissible because it presents irresolvable paradoxes and contradicts the very axioms of finite mathematics.  In recounting the history of the “actual infinite” (Hilbert’s designation of a C-infinity from Georg Cantor’s actual infinite set of numbers) Hilbert notes that the Russel-Zermelo paradox presents so many devastating contradictions that it nearly undermined deductive procedure in mathematics:
-These contradictions, the so-called paradoxes of set theory, though at first scattered, became progressively more acute and more serious.  In particular, a contradiction discovered by Zermelo and Russell had a downright catastrophic effect when it became known throughout the world of mathematics.  Confronted by these paradoxes, Dedekind and Frege completely abandoned their point of view [belief in the coherency of an infinite set as proposed by Cantor] and retreated.
+<blockquote>These contradictions, the so-called paradoxes of set theory, though at first scattered, became progressively more acute and more serious.  In particular, a contradiction discovered by Zermelo and Russell had a downright catastrophic effect when it became known throughout the world of mathematics.  Confronted by these paradoxes, Dedekind and Frege completely abandoned their point of view [belief in the coherency of an infinite set as proposed by Cantor] and retreated.<ref>Hilbert 1964, p. 141.</ref></blockquote>
 Hilbert then concludes that the technique of ideal elements (which can imply infinities) cannot be used if they change the fundamental axioms of finite numbers to which they have been applied.  Since this does not occur with potential infinities (B-infinities), but always occurs with actual infinities (C-infinities), Hilbert rejects the use of the latter in any way that could apply to the real world (i.e., real magnitudes, real counting, real series, etc.):

← Older revision	Revision as of 17:29, 16 July 2011
(No difference)

@@ Line 45: / Line 45: @@
 Henri Bergson wrestled with this problem, and finally made recourse to a kind of “protomentalist unified separation of existential states” which he termed “elementary memory.”  He supposed that this elementary memory existed in the universe as a whole, as a kind of very “elementary cosmic consciousness.”  In a famous passage in Duration and Simultaneity, he noted:
-What we wish to establish is that we cannot speak of a reality which endures without inserting consciousness into it.
+<blockquote>What we wish to establish is that we cannot speak of a reality which endures without inserting consciousness into it.<ref>Bergson 1965, p. 48. See also Spitzer 1989, pp. 12-14.</ref></blockquote>
 In order to show this, he constructs a thought experiment in which he assumes the above existential non-coincidence of incompatible states:
-We shall have to consider a moment in the unfolding of the universe, that is, a snapshot that exists independently of any consciousness, then we shall try conjointly to summon another moment brought as close as possible to the first, and thus have a minimum amount of time enter into the world without allowing the faintest glimmer of memory to go with it.  We shall see that this is impossible.  Without an elementary memory that connects the two moments, there will be only one or the other, consequently a single instant, no before and after, no succession, no time.
+<blockquote>We shall have to consider a moment in the unfolding of the universe, that is, a snapshot that exists independently of any consciousness, then we shall try conjointly to summon another moment brought as close as possible to the first, and thus have a minimum amount of time enter into the world without allowing the faintest glimmer of memory to go with it.  We shall see that this is impossible.  Without an elementary memory that connects the two moments, there will be only one or the other, consequently a single instant, no before and after, no succession, no time.<ref>Bergson 1965, p. 48.</ref></blockquote>
 I do not wish here to either affirm or deny Bergson’s protomentalist conclusions, but I do want to acknowledge the ontological conditions of change and time which Bergson recognized in concluding to them, namely,

@@ Line 33: / Line 33: @@
 This intrinsic analytical contradiction reveals the problematic character of the very idea of “infinite past time.”  It now remains for us to show the inapplicability of this problematic idea to our universe, and indeed, to any really possible changeable universe.  This step will give ontological (“synthetic”) significance to the analytical contradiction by showing that the condition of the real world (i.e., our real universe, or any really possible changeable universe) will contradict (and therefore resist) the application of this problematic idea to it.  The result will be that no real universe could have had infinite past time.
-Before we can proceed to this proof, we must first give an ontological explanation of real time , and then show that this real time must be intrinsic to any changeable universe, and then explain Hilbert’s distinction between actual and potential infinities so that it will be clear that “infinite past time” (as defined) must be an actual infinity which Hilbert shows to be inapplicable to any reality to which the axioms of finite mathematics can be applied.  The ontological proof against an infinity of past time will follow from this.
+Before we can proceed to this proof, we must first give an ontological explanation of real time<ref>“Real time” will be defined as “a continuous succession of non-contemporaneous distention".</ref>, and then show that this real time must be intrinsic to any changeable universe, and then explain Hilbert’s distinction between actual and potential infinities so that it will be clear that “infinite past time” (as defined) must be an actual infinity which Hilbert shows to be inapplicable to any reality to which the axioms of finite mathematics can be applied.  The ontological proof against an infinity of past time will follow from this.
 Please note that the a-priori synthetic proof is based upon the work of the famous mathematician David Hilbert (see “On the Infinite” in Hilary Putnam, ed Philosophy of Mathematics (Inglewood Cliffs, NJ: Prentiss Hall, 1977).  What follows is a brief summary of Hilbert’s discovery.  For a full explanation see NPEG, Chapter V, and PID Units 22 – 27.

@@ Line 1: / Line 1: @@
 MOERF – GOD UNIT G
-IS THERE EVIDENCE OF GOD FROM THE PHILOSOPHY OF MATHEMATICS?
+Is There Evidence of God From the Philosophy of Mathematics??
-© Robert J. Spitzer S.J./Magis Institute July 2011 – All rights reserved
+© Robert J. Spitzer, S.J. Ph.D./Magis Institute July 2011
 The foregoing is only the first step in the philosophical proof of a Creator of past time.  It shows that there is an analytical contradiction in the concept of “infinite past time” which strongly implies that this concept cannot exist in reality (and that past time must therefore be finite).  If readers are interested in the full a-priori synthetic proof for a beginning of past time (which necessitates a Creator of past time which itself is not in past time) then they may want to consult Chapter Five of NPEF and units 22 through 26 of PMD (see the Introduction to this book – Footnote #1).

← Older revision		Revision as of 01:05, 16 July 2011
Line 138:		Line 138:

	<references />		<references />
		+
		+	“Real time” will be defined as “a continuous succession of non-contemporaneous distension.”
		+
		+	Bergson 1965, p. 48. See also Spitzer 1989, pp. 12-14.
		+
		+	Bergson 1965, p. 48.
		+
		+	Beginning with Pseudo-Dionysius the Areopagite. See Rolt 1940.
		+
		+	Hilbert 1964, p. 135.
		+
		+	Hilbert 1964, p. 141.
		+
		+	Hilbert 1964, p. 151.
		+
		+	Craig 1993, p. 10. Many other mathematicians and philosophers have recognized this as well. See, for example, Whitrow, 1954-1955, pp. 215-225; 1967, pp. 422-432; 1968; and 1970, pp. 224-233.
		+
		+	Rotman and Kneebone 1966, p. 61. Italics mine.