865: Structural Mapping Theory/3
Mikhail Katz
katzmik at math.biu.ac.il
Fri Feb 12 07:01:49 EST 2021
This is a very intriguing text, and I spent some time perusing it though I
can't claim I read it all. I have a few problems with some of the
assumptions of your post.
1. Granted, you seek to present the position that ultimate foundations are
necessary. But how do you know that? It strikes me that in its
presentation here, the position is more of an ardently held opinion than a
convincing argument. Since this is a "meta-mathematical" (in the sense of
largely phiosophical) issue, it is disappointing that you did not engage
with any of the related work in the recent literature, some of which was
extensively debated at FoM.
2. I have, of course, encounted this type of opinion before. It strikes
me as a posture comparable to thinking that Euclidean space is primary when
you study geometry. I had a discussion with a colleague who is a
professional mathematician, who in fact linked both issues (set theory as
foundations of mathematics and Euclidean space as foundations for
geometry). Other mathematicians may also feel this way but as a
differential geometer I can tell you that my personal opinion is that such
a viewpoint (concerning Euclidean geometry) is naive and moreover untenable.
3. Admittedly a dilettante in foundations, I am puzzled by your
postulation of set theory as currently the only clear foundation. As you
probably know, many in category theory think that they possess clearer
foundations. It seems to me that stating your opinion more forcefully than
usual is not going to convince many people.
4. Some leading mathematicians working on the frontier of math and physics
(and I am thinking of Lou Kauffman in particular) expressed their
professional opinion that some of the entities they are dealing with are
just not sets. Without professional expertise in such fields, it seems
difficult to make sweeping claims about alleged ultimate foundations. I
don't recall you engaging Kauffman on the point he made. This of course
ties back to item 1 above.
M. Katz
On Thu, Feb 11, 2021 at 9:15 PM Harvey Friedman <hmflogic at gmail.com> wrote:
> SMAT (structural mapping theory) continues to clarify and evolve. The
> starting point that generates the excitement is
> [1]
> https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
> #113
> especially section 3.
>
> 1. Special Role of Set Theoretic Foundations - Ultimate Foundations
> 2. Many sorted structures.
> 3. Categories as many sorted structures
> 4. Building Structures out of Structures
>
> 1. SPECIAL ROLE OF SET THEORETIC FOUNDATIONS - ULTIMATE FOUNDATIONS
>
> Set theoretic foundations of mathematics is special in that it is
> based on a system of RIGID objects that do not gain their properties
> from contexts. This has well known drawbacks like, e.g., actually
> picking a specific set of counting numbers 0,1,2,... as von Neumann
> ordinals, something horrendously ghastly, artificial, repulsive, and
> wrong headed to most mathematicians. It is so bad that mathematics
> professors rather quickly disavow it and say that we know what they
> are and they are fluid and don't need that kind of definition, or that
> these numbers only exist in a context of a structure consisting of
> points with a relation among them, like a linearly ordered set where
> we only care about isomorphisms between such, etcetera. You know the
> drill. Of course some student raises their hand and asks what are
> these objects living in these structures, where do they come from? And
> the prof answers - we don't care, can come from anywhere. It nicely
> rhymes, doesn't it? And it you are uncomfortable with such a thing,
> then you don't really want to be a mathematician.
>
> I don't believe there is any kind of even remotely philosophically
> coherent ultimate foundational scheme that fails to be based on a
> fully rigid conceptual framework. Where entities have the properties
> they enjoy in light of what exactly they are, not the context in which
> they are in. This ultimate foundation is, at least today, known as set
> theory, with its rigid (intended to be) completely objective entities.
> There is an arguably problematic extension to class theory, and this
> raises a lot of unresolved questions in my mind. I'm not quite sure
> what my attitude towards class theory is at this point, but there is
> no question that it does offer some rather striking advantages laying
> ultimate foundations for certain structures as proper classes which
> are somehow quite a bit simpler to conceptualize than putting
> cardinality restrictions down. That's a complex topic for another day.
>
> So set theory starts with say emptyset and {emptyset} and this tells
> us a lot. These are the most incredibly boring and lifeless
> mathematical objects you can think of, enough to put a dragon to
> sleep. Yet their purity, their, objectivity, their clarity, and their
> rigidity, is really something to revel in.
>
> What really makes set theoretic foundations so promising as the
> ultimate foundations is the notion of hereditarily finite set - this
> is the result of continuing from this boring beginning emptyset,
> {emptyset}, by combining sets. Most conveniently, by taking singletons
> and pairwise unions. This generates a remarkably elegant and powerful
> universe of sets - the hereditarily finite sets - that have remarkably
> simple and powerful properties, a tiny number of which FORMALLY DERIVE
> the others. There is a lot more to be said about this, and I have said
> some elsewhere, but will revisit it at a later posting.
>
> Of course, as we know, when we really take the lid off, and throw away
> any limiting axiom (that forces every set to be "finite"), then we are
> into general infinitary set theory, and NO LONGER IS IT THE CASE that
> a tiny number of principles logically derives the rest. E.g., power
> set and union aren't derivable from more basic principles, as THEY ARE
> in the hereditarily finite universe we have been talking about. BUT
> there are some striking ways in which ZF and other systems are built
> canonically from such finite set theories. These creates a rather
> incredibly satisfying powerful edifice for an ULTIMATE FOUNDATION.
>
> NOW NOBODY wants to have mathematics do away with its widely use
> categorical language and its special attention to mathematical objects
> requiring contexts to be useful and emphasizing the relationships
> between structures rather than individuals in isolation and so forth.
> But IT MUST BE DISCIPLINED by ultimate foundations.
>
> Fortunately, almost entirely, this is fairly straightforward to
> achieve, at least through the stuff generated by Eilenberg and Mac
> Lane. For some specifics that we will need, see 4, although that will
> be in a later posting. The situation seems to be far less clear cut,
> and rather murky, with some more radical ideas that try to really try
> to do away with the usual distinction between isomorphic and
> identical. Such ideas must be discipline by our Ultimate Foundations.
> Any attempt to circumvent that process requires a major
> PHILOSOPHICALLY COHERENT development that needs to be fully fleshed
> out and scrutinized.
>
> 2. MANY SORTED STRUCTURES
>
> There is a very natural useful and well studied concept of "structure"
> that is superbly flexible and workable coming naturally out of
> Ultimate Foundations. This is Many Sorted Structures. It's not that
> they are part of the Ultimate Axioms we use. It's just that they are
> extremely natural and directly usable.
>
> Actually not quite. With say real numbers, we have division, and
> division by zero is something that our mothers told us not to do. So
> what is 1/0? The straightforward answer is that it is undefined. This
> leads to what is called FREE LOGIC which allows undefined terms. There
> are a few brands of this and some of them involve subtle philosophical
> issues about the nature of existence and reference and so forth. I AM
> NOT TALKING ABOUT ANY OF THAT HERE. I am thinking of a very
> straightforward well worked out ISSUELESS version of FREE LOGIC. It
> has obvious foundations.
>
> It uses uparrows for "undefined" and downarrows for "defined" and =
> for "both sides are defined and the same" and wigglequals for "both
> sides are either = or both sides are undefined". The uparrows and the
> downarrows are used as suffixes to terms and form statements. It is
> required that constants are always defined. Terms are defined if and
> only if ALL of its subterms are defined. Relations at terms always
> make sense, even if some of those terms are not defined. If at least
> one of the terms is undefined, then the relation is automatically
> considered to be FALSE at those terms. Quantification is normal and
> quantifies over existing objects (what else could there be?).
>
> So e.g.,
> 1/0 uparrows is true
> 1/0 downarrows is false
> 1/0 = 1/0 is false
> 1/0 wigglequals 1/0 is true
> 1/0 < 1/0 is false
> 1/0 <= 1/0 is false
> -1 < 1/0 is false
>
> The nice extension of predicate logic for this is well known, with
> full syntax and semantics and completeness of Hilbert and Gentzen type
> systems (references????).
>
> A many sorted structure consists of at least one and at most finitely
> many named sorts, each of which are sets. (The class possibility is
> important and will be discussed later). There are zero or more, but
> only finitely many, named constants, named relations, and named
> functions. Equality on each sort is taken as given, but not across
> sorts. Sorts may overlap but that is not reflected in the syntax.
>
> Each named constant is drawn from a sort and that sort is part of its
> name. Of course it may be an element of several of the sorts, but one
> has to be picked for its name.
>
> Each named relation has an arity k >= 1, and is on a list of k sorts.
> It has to be a subset of the Cartesian product of these k sorts. k and
> the names of those k+1 sorts are part of the name of the relation.
>
> Each named function has an arity k >= 1, and is from a list of k sorts
> into a single sort. It has to be a partial function from the Cartesian
> product of the k sorts into the single sort.
>
>
>
>
>
>
>
>
>
> I give a simple argument below that set theory is SUPREME in the sense
> that it demands that everything else be ultimately interpreted in set
> theory IN THE BACKGROUND, but not in any practical sense of course.
> Normally there set theorists will have no issues with what you do, and
> probably admire it, and feel comfortable letting you work on a leash,
> provided there are no issues about how to generally routinely
> interpret what you are doing in the standard ways set theoretically.
> HOWEVER, once that becomes murky, then a big issue arises that needs
> to be addressed. And the SUPREME COURT is set theory. Our attitude
> towards INSURRECTIONS is actually rather positie. We seek not to
> imprison you, or throw you into internment camps, but rather HELP YOU
> SEE how to hopefully routinely adjust what you are doing so that it is
> clear there are no longer any issues to give a proper set theoretic
> interpretation. Sometimes this requires the experience and expertise
> of a standard classical foundations expert who knows the incredible
> sometimes hidden power of set theory to do all sorts of things its
> founders never considered. Statements by various cultural heros, dead
> or alive, that set theory foundations is insufficient for this or that
> or cannot forge tools naturally to do this or that are HIGHLY SUSPECT,
> and generally come from people not properly schooled in many decades
> of classical set theoretic foundational thought.
>
> The reason that set theory is THE SUPREME COMMANDER, and THE SUPREME
> COURT, is that it is founded on RIGIDITY. The emptyset and say
> {emptyset} and the like, the most lifeless boring things imaginable,
> have an immutable unchanging forever existence where my emptyset and
> my {emptyset} are the same as yours, and your newborn baby's also. If
> my roof collapses or I get fired or a get a promotion, then my
> emptyset and {emptyset} is not going to be affected at all. There is
> no RELATIVITY in set theory.
>
> There is a mathematical fact that is at the heat of the support for
> this RIGIDITY. And this is that for sets under membership, all
> automorphisms are the identity. Yes I know this makes things go
> against the grain of a lot of practical mathematical thinking that
> facilitates our making mathematical advances such as no x^n + y^n =
> z^n, n >= 3. No major problem in interpreting that fact set
> theoretically, but things get seriously anti set theoretic in spirit
> when rigorously documenting the proof. That's fine with the SUPREMES.
> If that facilitates real mathematical progress and rigor, then all the
> better. But don't forget who the SUPREMES are.
>
> This RIGIDITY has spawned a way of thinking of its own, quite
> different than what core mathematics relishes in, that has given us
> INCREDIBLE FOUNDATIONAL ADVANCES of the GREATEST GENERAL INTELLECTUAL
> INTEREST. And probably only VERY CLUMSY hints of what is to come even
> by 2100, a blink of an eye. It is preposterous to believe that these
> VERY CLUMSY but incredible advances would be at all possible without
> the vast technologies that have been developed from the RIGID POINT OF
> VIEW.
>
> 2. MANY SORTED RELATIONAL STRUCTURES
>
> The SUPREME COURT has ruled that all structures are to ultimately be
> interpreted as a MANY SORTED RELATIONAL STRUCTURE. The Supremes have
> not yet ruled on the validity of the notion of class, generally
> considered to be set like entities which are too big to be sets.
> Proper classes are flagged as under investigation, something in limbo.
> There have been no rulings against them. The Court much prefers you
> pick a suitable cardinal lambda and work within the cumulative
> hierarchy below lambda rather than think you are working with all
> sets. The Supremes also urge you to consider using large cardinal
> hypotheses, of course labeling them as such, as they seem to be
> becoming so useful and INHERENTLY NECESSARY in a growing list of basic
> normal type rich situations.
>
> Many sorted relational structures some with one or more sorts, but at
> most finitely many sorts. The sorts have names, and are sets, possibly
> overlapping. There are also named constants, relations, and functions.
> There are finitely many of these too. Each constant is given a unique
> sort and is an element of that sort. Even if it is an element of more
> than one sort, it is still assigned a particular sort. The named
> relations are assigned an arity n, and n sorts S_1,...,S_n. R is
> assigned a subset of S_1 x ... x S_n. The named functions are also
> assigned an arity n, and n+1 sorts S_1,...,S_n+1. F is assigned a
> function from S_1 x ... x S_n into S_n+1.
>
> The signature of a many sorted structure consists of its shape, which
> is the list of named sorts, the list of named constants and their
> assigned sorts, the list of named relations and their assigned sorts
> for their arguments, and the list of named functions and their
> assigned sorts for their arguments and values. Thus the signature is a
> finite object.
>
> For each sort S there are variables v_i^S, i >= 1, that range over S.
> Terms and their sorts are built up by induction. The atomic formulas
> are the equations between terms of the same sort, and primitive
> relations at the right number of terms of the right sorts. Formulas
> are built up in the usual way through the connectives not, and, or, if
> then, if and only if, and quantification over each sort.
>
> ##########################################
>
> 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 865th 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 #1-799 can be found at
> http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
>
> 800: Beyond Perfectly Natural/6 4/3/18 8:37PM
> 801: Big Foundational Issues/1 4/4/18 12:15AM
> 802: Systematic f.o.m./1 4/4/18 1:06AM
> 803: Perfectly Natural/7 4/11/18 1:02AM
> 804: Beyond Perfectly Natural/8 4/12/18 11:23PM
> 805: Beyond Perfectly Natural/9 4/20/18 10:47PM
> 806: Beyond Perfectly Natural/10 4/22/18 9:06PM
> 807: Beyond Perfectly Natural/11 4/29/18 9:19PM
> 808: Big Foundational Issues/2 5/1/18 12:24AM
> 809: Goedel's Second Reworked/1 5/20/18 3:47PM
> 810: Goedel's Second Reworked/2 5/23/18 10:59AM
> 811: Big Foundational Issues/3 5/23/18 10:06PM
> 812: Goedel's Second Reworked/3 5/24/18 9:57AM
> 813: Beyond Perfectly Natural/12 05/29/18 6:22AM
> 814: Beyond Perfectly Natural/13 6/3/18 2:05PM
> 815: Beyond Perfectly Natural/14 6/5/18 9:41PM
> 816: Beyond Perfectly Natural/15 6/8/18 1:20AM
> 817: Beyond Perfectly Natural/16 Jun 13 01:08:40
> 818: Beyond Perfectly Natural/17 6/13/18 4:16PM
> 819: Sugared ZFC Formalization/1 6/13/18 6:42PM
> 820: Sugared ZFC Formalization/2 6/14/18 6:45PM
> 821: Beyond Perfectly Natural/18 6/17/18 1:11AM
> 822: Tangible Incompleteness/1 7/14/18 10:56PM
> 823: Tangible Incompleteness/2 7/17/18 10:54PM
> 824: Tangible Incompleteness/3 7/18/18 11:13PM
> 825: Tangible Incompleteness/4 7/20/18 12:37AM
> 826: Tangible Incompleteness/5 7/26/18 11:37PM
> 827: Tangible Incompleteness Restarted/1 9/23/19 11:19PM
> 828: Tangible Incompleteness Restarted/2 9/23/19 11:19PM
> 829: Tangible Incompleteness Restarted/3 9/23/19 11:20PM
> 830: Tangible Incompleteness Restarted/4 9/26/19 1:17 PM
> 831: Tangible Incompleteness Restarted/5 9/29/19 2:54AM
> 832: Tangible Incompleteness Restarted/6 10/2/19 1:15PM
> 833: Tangible Incompleteness Restarted/7 10/5/19 2:34PM
> 834: Tangible Incompleteness Restarted/8 10/10/19 5:02PM
> 835: Tangible Incompleteness Restarted/9 10/13/19 4:50AM
> 836: Tangible Incompleteness Restarted/10 10/14/19 12:34PM
> 837: Tangible Incompleteness Restarted/11 10/18/20 02:58AM
> 838: New Tangible Incompleteness/1 1/11/20 1:04PM
> 839: New Tangible Incompleteness/2 1/13/20 1:10 PM
> 840: New Tangible Incompleteness/3 1/14/20 4:50PM
> 841: New Tangible Incompleteness/4 1/15/20 1:58PM
> 842: Gromov's "most powerful language" and set theory 2/8/20 2:53AM
> 843: Brand New Tangible Incompleteness/1 3/22/20 10:50PM
> 844: Brand New Tangible Incompleteness/2 3/24/20 12:37AM
> 845: Brand New Tangible Incompleteness/3 3/28/20 7:25AM
> 846: Brand New Tangible Incompleteness/4 4/1/20 12:32 AM
> 847: Brand New Tangible Incompleteness/5 4/9/20 1 34AM
> 848. Set Equation Theory/1 4/15 11:45PM
> 849. Set Equation Theory/2 4/16/20 4:50PM
> 850: Set Equation Theory/3 4/26/20 12:06AM
> 851: Product Inequality Theory/1 4/29/20 12:08AM
> 852: Order Theoretic Maximality/1 4/30/20 7:17PM
> 853: Embedded Maximality (revisited)/1 5/3/20 10:19PM
> 854: Lower R Invariant Maximal Sets/1: 5/14/20 11:32PM
> 855: Lower Equivalent and Stable Maximal Sets/1 5/17/20 4:25PM
> 856: Finite Increasing reducers/1 6/18/20 4 17PM :
> 857: Finite Increasing reducers/2 6/16/20 6:30PM
> 858: Mathematical Representations of Ordinals/1 6/18/20 3:30AM
> 859. Incompleteness by Effectivization/1 6/19/20 1132PM :
> 860: Unary Regressive Growth/1 8/120 9:50PM
> 861: Simplified Axioms for Class Theory 9/16/20 9:17PM
> 862: Symmetric Semigroups 2/2/21 9:11 PM
> 863: Structural Mapping Theory/1 2/4/21 11:36PM
> 864: Structural Mapping Theory/2 2/7/21 1:07AM
>
> Harvey Friedman
>
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