[FOM] 729: Consistency of Mathematics/1

[Mghanga Mwakima]

Related to 1.), I think is an epistemological question about mathematics,
which is roughly: how do we know the truths of mathematics? One approach
would be to say mathematical knowledge is an instance of apriori knowledge.
But then one could still demand for the grounds of the apriori
justification that mathematics enjoys. Here, it one could say that the
apriori justification is to be found in the analyticity of mathematics. But
is there a consensus on what analyticity in mathematics is?

Related to this and your last point about philosophical geometry. If one
distinguishes between physical (applied) geometry and pure geometry, then
one could say that the former (applied geometry) is not apriori, hence not
analytic, because it depends on the physical structure of spacetime. But
how would one characterise pure geometry? Is: 1.) pure geometry analytic
and 2.) in what sense? Those who want to reject a "devoid of factual
content" characterization of the analyticity, what exactly constitutes the
"factual content" or "mathematical content" of pure geometry that would
make one reject this characterization for pure geometry as well?

In discussions I have had, I get the impression that pure (Euclidean)
geometry is unlike a theory like ZFC for two reasons: 1.) because by a
result of Tarski, the former is complete. And 2.) ZFC is unlike pure
geometry because ZFC has "mathematical content" in the sense that it it
implies the existence of very rich (perhaps infinite) mathematical
structures. This is one reason why we might demur applying the sense of
"devoid of factual content" to the analyticity of ZFC.

In short, 1.) what do we mean when we say that certain theories have
"factual content"?
and 2.) Given the answer to 1.) how would we characterize the analyticity
(or lack thereof) in an appropriate sense of pure geometry, (say Euclidean)
and ZFC?

On Tue, Oct 25, 2016 at 1:25 PM, Harvey Friedman <hmflogic at gmail.com> wrote:

> What are the most monumental issues in the foundations of mathematics?
> It would be valuable to hear what subscribers think, but the two
> obvious classical ones are, in no particularly order:
>
> 1. Is mathematics consistent - I.e., free of contradiction? After all,
> it deals with objects not accessible to us by observations in any
> usual sense. Not only does mathematics use infinitely many objects, it
> uses myriads of completed infinite totalities. Can we, and how can we
> be assured of the consistency of mathematics? What would constitute
> evidence or confirmation of the consistency of mathematics?
>
> 2. Is the apparent overwhelming power of the usual foundations for
> mathematics through ZFC of any real use in ordinary mathematical
> environments? What if anything is there to gain from extending ZFC in
> terms of ordinary mathematical environments? Or is there some kind of
> completeness of ZFC for typical problems arising from familiar
> concrete mathematical investigations?
>
> I have worked and written the most on 2, under Concrete Mathematical
> Incompleteness, and this has culminated in the series on Emulation
> Theory, titled Large Cardinals and Emulations/1-25 and continuing.
> This adventure has taken on the general character of Musical
> Composition in the following sense. Works of musical composition of
> course do not normally conform to specific previously stated problems
> and challenges. But the masterpieces exude a compelling
> *inevitability*, a critical aspect of Musical Composition that is
> discussed so eloquently in, e.g.,
> https://www.awesomestories.com/asset/view/Bernstein-
> Explains-Beethoven-s-Fifth-Part-1
> https://www.youtube.com/watch?v=KI1klmXUER8 See especially 8:20-37,
> and 24:00-40. I view myself as searching for an apparently related
> kind of *inevitability* in Concrete Mathematical Incompleteness.
>
> For those who take abstract set theory as intrinsically fundamentally
> important in its own right, independently of the quality and quantity
> of its connections with the rest of mathematics, on this list would be
>
> 3. Is the continuum hypothesis true or false? Can we and how can we
> obtain evidence or confirmation either way?
>
> Research on 3 could use a refreshing new methodology, which I hope to
> be offering in Refuting the Continuum Hypothesis?/1-9 and hopefully
> continuing.
>
> There are some other important issues in f.o.m., including ones that I
> would regard as less than monumental; it would be interesting to hear
> from subscribers about their views.
>
> There are some monumental, or arguably monumental issues that I would
> not classify as foundations of mathematics, bur rather foundations of
> computer science. This is way beyond the scope of this series, and so
> I will just briefly mention a) the notorious complexity issues
> everybody is aware of, and b) Church's Thesis. Actually b) is such a
> radically different problem than a), that I prefer to view b) as an
> arguably monumental issue in Philosophical Computer Science, which I
> like to distinguish from Foundations of Computer Science, and
> certainly from Computer Science itself. So perhaps we can add
> Philosophical Computer Science to a list of subjects that includes
> Philosophical lGeometry - although for Philosophical Geometry I do
> have a reasonably coherent and perhaps resonating view as to just what
> I mean.
>
> So I have seriously thought about 1,2,3 and b), and written about them
> on the FOM. With regard to a), I do not admit to have worked on it.
>
> In this series I want to deal only with 1. I have written about 70
> pieces about and surrounding 1, including these:
>
> FROM https://u.osu.edu/friedman.8/foundational-adventures/
> downloadable-manuscripts/
>
> #10. The interpretation of set theory in pure predication theory,
> August 20, 1997, 64 pages.
> #12. The interpretation of set theory in mathematical predication
> theory, October 25, 1997, 10 pages.
> #13. The axiomatization of set theory by extensionality, separation,
> and reducibility, October 28, 1997, 71 pages
> #35. A Way Out, August 28, 2002, 49 pages. Also in: A Way Out, in: One
> Hundred Years of Russell’s Paradox, ed. Godeharad Link, de Gruyter,
> 49-86, 2004.
> #39. Sentential Reflection, January 5, 2003, 3 pages,
> #40. Restrictions and extensions, February 17, 2003, 3 pages.
> #41. Elemental sentential reflection, March 3, 2003, 6 pages.
> #42. Similar subclasses, March 11, 2003, 3 pages.
> #46. Relational system theory, May 26, 2005, 24 pages.
> #52. Concept Calculus, October 25, 2006, 42 pages,
> #59. 3. Interpreting Set Theory in Ordinary Thinking: Concept
> Calculus, 12 pages.
> #62. Concept Calculus: Much Better Than, October 31, 2009, 48 pages.
> Also in: Concept Calculus: Much Better Than, in: New Frontiers in
> Research on Infinity, ed. Michael Heller and W. Hugh Woodin, Cambridge
> University Press, 130-164, 2011.
> #72. Concept Calculus: universes. October 2, 2012, 33 pages,
> #74. A Divine Consistency Proof for Mathematics. December 25, 2012, 70
> pages.
> #82. Testing the Consistency of Mathematics, 7 pages, July 23, 2014.
> #83. Flat Mental Pictures, September 5, 2014, 5 pages.
> #85. Conservative Growth: A Unified Approach to Logical Strength,
> October 12, 2014, 15 pages.
>
> FROM https://u.osu.edu/friedman.8/foundational-adventures/
> downloadable-lecture-notes-2/
>
> #35. Concept Calculus, Mathematical Methods in Philosophy, Banff,
> Canada, February 21, 2007, 9 pages.
> #37. Concept Calculus’ APA Panel on Logic in Philosophy, APA Eastern
> Division Annual Meeting, Baltimore Maryland, January 2, 2008, 17
> pages.
> #39. Concept Calculus, Carnegie Mellon University, March 26, 2009,
> Pure and Applied Logic Colloquium
> #43. Concept Calculus, Department of Philosophy, MIT, November 4,
> 2009, 22 pages.
> #61. Concept Calculus, Institutional Honorary Doctorate, Ghent
> University, Ghent, Belgium, September 5, 2013,  26 pages.
>
> FOM POSTINGS. See FOM Archives, http://www.cs.nyu.edu/pipermail/fom/
>
> #41: Strong Philosophical Indiscernibles
> #90: Two Universes 6/23/00 1:34PM
> #110: Communicating Minds I 12/19/01 1:27PM
> #111: Communicating Minds II 12/22/01 8:28AM
> #112: Communicating MInds III 12/23/01 8:11PM
> #116: Communicating Minds IV 1/4/02 2:02AM
> #122: Communicating Minds IV-revised 1/31/02 2:48PM
> #155: A way out  8/13/02  6:56PM Also in: A Way Out, in: One Hundred
> Years of Russell’s Paradox, ed. Godeharad Link, de Gruyter, 49-86,
> 2004.
> #156: Societies  8/13/02  6:56PM
> #157: Finite Societies  8/13/02  6:56PM
> #158: Sentential Reflection  3/31/03  12:17AM
> #159. Elemental Sentential Reflection  3/31/03  12:17AM
> #160. Similar Subclasses  3/31/03  12:17AM
> #161: Restrictions and Extensions  3/31/03  12:18AM
> #200: Advances in Sentential Reflection 12/22/03 11:17PM
> #245: Relational System Theory 1  5/16/05  12:24PM
> #246: Relational System Theory 2  5/15/05  9:57PM
> #248: Relational System Theory 2/restated  5/26/05  1:46AM
> #256. NAME:finite inclusion theory  11/21/05  2:34AM
> #257. FIT/more  11/22/05  5:34AM
> #283: A theory of indiscernibles  5/7/06  6:42PM
> #290: Independently Free Minds/Collectively Random Agents 6/12/06
> 11:01AM
> #291: Independently Free Minds/Collectively Random Agents (more) 6/13/06
> 5:01PM
> #292: Concept Calculus 1  6/17/06  5:26PM
> #293: Concept Calculus 2  6/20/06  6:27PM
> #294: Concept Calculus 3  6/25/06  5:15PM
> #295: Concept Calculus 4  7/3/06  2:34AM
> #298: Concept Calculus 5  7/14/06  5:40AM
> #444: The Exploding Universe 1  11/1/10  1:46AM
> #509: A Divine Consistency Proof for Mathematics  12/26/12  2:15AM
> #511: A Supernatural Consistency Proof for Mathematics   1/10/13  9:19PM
> #513: Five Supernatural Consistency Proofs for Mathematics  1/14/13  1:13AM
> #515: Eight Supernatural Consistency Proofs For Mathematics  1/19/13
> 2:40PM
> #524: Testing Consistency of Math  7/23/14  6:49AM
> #527: Some Mental Pictures i  8/11/14  11:13AM
> #547: Conservative Growth – Triples  9/29/14  11:34PM
> #549: Conservative Growth – beyond triples  10/6/14  1:31AM
> #556: Flat Foundations 1  10/29/14  4:07PM
>
> SUMMARY OF THE ABOVE
>
> Given the number of these items, it is important for me to give a
> guide as to what is here.
>
> #82 https://u.osu.edu/friedman.8/foundational-adventures/
> downloadable-manuscripts/
> and FOM posting #524 are in a category by themselves, as they do not
> provide a consistency proof of ZFC and large cardinal extensions, but
> rather discusses computer based confirmation of such consistency. In
> this series, we will be updating these to relate more closely to
> Emulation Theory.
>
> #74 https://u.osu.edu/friedman.8/foundational-adventures/
> downloadable-manuscripts/
> and arguably FOM postings #509, #511, #513, #515, are also in a
> category by themselves, as they specifically use axioms directly
> motivated by classic writings in Theology. The first of these is at
> roughly at the level of a measurable cardinal (a little below), and
> the four FOM postings are at various levels, below and above a
> measurable cardinal. More specifically, the first of these is based on
> the concept of positive/negative property (good to have the positive
> side, bad to have the negative side), used explicitly by Goedel, and
> at least implicitly by Leibniz and St. Anselm. The other four use a
> distinction between real and transcendental objects, and also a
> mapping from real objects to generally transcendental objects as in
> Doppelgängers - https://en.wikipedia.org/wiki/Doppelgänger
>
> A principal theme in the previous paragraph and all of the other
> approximately 70 pieces cited is "flatness". The opposite of flatness
> is of course the cumulative hierarchy of sets. A typical flat setup is
> to have a sort for objects (none of which are sets) and sets of
> objects - but if we just use objects and sets of objects we generally
> want a pairing function on objects. Or at least a sort for objects and
> a sort of binary (or higher arity) relations on objects. There can of
> course be more primitives, and maybe a sort for one higher type.
>
> There is a group of pieces titled Concept Calculus where the
> primitives are motivated by naive physics of space and time. The
> driving intuitive principle is that "anything that can happen will
> happen" or "any configuration that is possible is realized". This idea
> goes back to Aristotle with Plenitude. (There are some other
> directions titled Concept Calculus that are not based on the naive
> physics of space and time - see below.) For those that are based on
> the naive physics of space and time, see
> #52, #72, https://u.osu.edu/friedman.8/foundational-adventures/
> downloadable-manuscripts/
> everything under
> https://u.osu.edu/friedman.8/foundational-adventures/
> downloadable-lecture-notes-2/
> above
> #292 6/17/06 #293 6/20/06 #294: 6/25/06 #295: 7/3/06 #298: 7/14/06
> #444: 11/1/10, http://www.cs.nyu.edu/pipermail/fom/
>
> There are some pieces titled Concept Calculus that are not motivated
> by naive physics of space and time. They use the primitives better
> than and much better than, an example of concept amplification.
> #52, #62, https://u.osu.edu/friedman.8/foundational-adventures/
> downloadable-manuscripts/
> everything under
> https://u.osu.edu/friedman.8/foundational-adventures/
> downloadable-lecture-notes-2/
> above
> #292 6/17/06 #293 6/20/06 #294: 6/25/06 #295: 7/3/06 #298: 7/14/06,
> http://www.cs.nyu.edu/pipermail/fom/
> I think of "much better than" as an example of concept amplification.
>
> There is the A Way Out approach to Russell's Paradox, 335,
> https://u.osu.edu/friedman.8/foundational-adventures/
> downloadable-manuscripts/
> and #155, http://www.cs.nyu.edu/pipermail/fom/ . This is a direct
> natural weakening of the full comprehension scheme that takes the
> form: Every virtual set forms a set, or, outside any given set, has
> two inequivalent elements, where all elements of the virtual set
> belonging to the first belong
> to the second.The resulting formal system is mutually interpretable
> with certain large cardinals.
>
> There are some pieces that focus on two minds. The basic idea is that
> the first mind is dominated by the second mind, which has considerably
> more objects than the first mind. However, they have a fundamental
> compatibility that allows them to communicate effectively: they agree
> on (certain) statements that involve only the objects of the first
> mind. This leads to associated formal systems of various logical
> strengths some going to extremely large cardinals. The ones about
> societies are different, and are focused on the "like" relation among
> people, which leads to some strong theories that are natural under the
> "like" interpretation, motivated by the A Way Out approach.
> #110: 12/19/01 #111: 12/22/01 #112: 12/23/01 #116: 1/4/02 #122:
> 1/31/02 #156: 8/13/02 #157: 8/13/02 #290: 6/12/06 #291: 6/13/06,
> http://www.cs.nyu.edu/pipermail/fom/
>
> There are some pieces that provide a kind of unified thematic approach
> that climbs from the logically weak to the logically very strong.
> These include
> #83, #85, https://u.osu.edu/friedman.8/foundational-adventures/
> downloadable-manuscripts/
> #527: 8/11/14  #547: 9/29/14 #549: 10/6/14 #556: 10/29/14
> http://www.cs.nyu.edu/pipermail/fom/
>
> There are some other pieces not included in this summary, but I want
> to stop here.
>
> In the next posting, we will start with revisiting the Communicating
> Minds approach. I now think of it as Expanding Minds. Also I will
> consolidate it with the Real/Transcendental approach connected with
> theology.
>
> ***********************************************
> My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
> https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
> This is the 729th in a series of self contained numbered
> postings to FOM covering a wide range of topics in f.o.m. The list of
> previous numbered postings #1-699 can be found at
> http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
>
> 700: Large Cardinals and Continuations/14  8/1/16  11:01AM
> 701: Extending Functions/1  8/10/16  10:02AM
> 702: Large Cardinals and Continuations/15  8/22/16  9:22PM
> 703: Large Cardinals and Continuations/16  8/26/16  12:03AM
> 704: Large Cardinals and Continuations/17  8/31/16  12:55AM
> 705: Large Cardinals and Continuations/18  8/31/16  11:47PM
> 706: Second Incompleteness/1  7/5/16  2:03AM
> 707: Second Incompleteness/2  9/8/16  3:37PM
> 708: Second Incompleteness/3  9/11/16  10:33PM
> 709: Large Cardinals and Continuations/19  9/13/16 4:17AM
> 710: Large Cardinals and Continuations/20  9/14/16  1:27AM
> 711: Large Cardinals and Continuations/21  9/18/16 10:42AM
> 712: PA Incompleteness/1  9/2316  1:20AM
> 713: Foundations of Geometry/1  9/24/16  2:09PM
> 714: Foundations of Geometry/2  9/25/16  10:26PM
> 715: Foundations of Geometry/3  9/27/16  1:08AM
> 716: Foundations of Geometry/4  9/27/16  10:25PM
> 717: Foundations of Geometry/5  9/30/16  12:16AM
> 718: Foundations of Geometry/6  101/16  12:19PM
> 719: Large Cardinals and Emulations/22
> 720: Foundations of Geometry/7  10/2/16  1:59PM
> 721: Large Cardinals and Emulations//23  10/4/16  2:35AM
> 722: Large Cardinals and Emulations/24  10/616  1:59AM
> 723: Philosophical Geometry/8  10/816  1:47AM
> 724: Philosophical Geometry/9  10/10/16  9:36AM
> 725: Philosophical Geometry/10  10/14/16  10:16PM
> 726: Philosophical Geometry/11  10/17/16  4:04PM
> 727: Large Cardinals and Emulations/25  10/20/16  1:37PM
> 728: Philosophical Geometry/12  10/24/16  3:35PM
>
> Harvey Friedman
> _______________________________________________
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> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>



-- 
Mghanga David Mwakima
Cell: +1-413-331-8718
Google Voice: +1-617-902-0478
Email: mwakima.david at gmail.com
"I have never let my schooling interfere with my education." Mark Twain?
Grant Allen?
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