Pluralism in the Ontology of Mathematics

[Doug Mounce]

Interesting talk by Joel David Hamkins.  About two-thirds of the way through he discusses how the direction of research could have gone two ways after the power of set theory was uncovered.

http://jdh.hamkins.org/pluralism-in-the-ontology-of-mathematics-mamuphi-paris-february-2022/

Abstract: What is the nature of mathematical ontology—what does it mean to make existence assertions in mathematics? Is there an ideal mathematical realm, a mathematical universe, that those assertions are about? Perhaps there is more than one. Does every mathematical assertion ultimately have a definitive truth value? I shall lay out some of the back-and-forth in what is currently a vigorous debate taking place in the philosophy of set theory concerning pluralism in the set-theoretic foundations, concerning whether there is just one set-theoretic universe underlying our mathematical claims or whether there is a diversity of possible set-theoretic conceptions.

~/

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[David Bibby]

Dear Doug,

I agree, very interesting talk, thanks.  I quite liked the dots along the top which indicated the progress through the slides!

There was one slide which described the cumulative universe as a vast hierarchy of sets, sets of sets, sets of sets of sets … founded ultimately upon the empty set.  Contrast that with Lonergan’s explanation of living species in the development within genetic method:

“The fundamental element in emergent probability is the conditioned series of things and schemes; that series is realised cumulatively in accord with successive schedules of probabilities; but a species is not conceived as an accumulated aggregate of theoretically observable variations; on the contrary, it is an intelligible solution to a problem of living in a given environment, where the living is a higher systematisation of a controlled aggregation of aggregates of aggregates of aggregates, and the environment tends to be constituted more and more by other living things.”  (Insight, chapter 8, section 6, 2008/290)

I wonder whether we could form a set theoretic conception of that higher systematisation of aggregates of aggregates of aggregates…  The mathematical cumulative universe is built upon the empty set, but nothing concrete can be built from nothing.  The mathematical universe is abstract, ideal.  Within that abstract universe, there could be a pluralism of ontologies (as suggested by the speaker), but there was no mention of insight to pivot between the abstract and the concrete, to give it a sense of the real.  Knowledge of the real only arises when there is a complex, intelligent, species, able to make virtually unconditioned judgements by abstracting from the empirical residue of the universe in which it finds itself.  In other words, the epistemological question cannot be settled outside, but only within the set theoretic structure itself, when it describes a living, intelligent, autonomous, species, and genetic method is the best answer that species can give within the limitations of the coincidental aggregates of its experience.

Kind regards,

David

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[Doug Mounce]

Hi David,

I think Hamkins did say that just-after set theory began to prove useful then research could have gone in two directions - either in empirical application to existing problems or in proliferation of speculative set theoretic possibilities, and researchers by and large chose the latter. 

I would guess that some concepts in Lonergan's work could be construed in set theoretic terms; the definition of an event, for example.

"What, then, is an event?"  Lonergan asks.  "The simplest answer is to say that it is a primitive notion too simple and obvious to be explained."  Of course, he goes-on to explain what is meant by, "the puzzling name `event'", and aggregate sets certainly play a role.

PS - if I can dovetail on your discussion of math then I'm inclined to begin with formal systems (i.e., what you can teach a computer - like arithmetic or chess) with semantic translation such that arithmetic informs us about numbers or geometry about space.  Logic is a form of math, in that definition, and Lonergan has the insight about how axioms must be continually added to shore up complete systems of logic.  FWIW

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[jaraymaker]

 

Doug, David,

 

I do have a question:  how can the whole of math be "ultimately" founded upon the empty set in the light of the below (the first half of which comes from Wikipedia as I googled "empty set")?

1) In math, the empty set is the unique set having no elements; its size (count of elements in a set) is zero. Some axiomatitc set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.

Any set other than the empty set is called non-empty.

In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). The empty set may also be called the void set. 

 

2) George Boole introduced the concept of empty set, or "nothing" as he called it, as the complement to the "universe" in his Mathematical Analysis of Logic (1847). His notations for them were somewhat boring, 0 and 1 respectively. Cantor wrote in 1880 "for the absence of points we choose the letter O".  End of quotes         John

 

[jaraymaker]

Hi all,

 

or to put my question in a different way: are the 0's and 1's introduced  by Boole as well as his CONCEPT of empty as "foundation" and "complement to the universe",  is not these profound insights of his not the "foundation" of the pragmatic digitalization now driving the world and making ignore the true, deeper foundation on which Lonergan built his method? 

 

That is, the empty set seems to me to be one form of the already already-out-there-now-real  while neglecting or downplaying things, which exist as intelligible unity-identity-wholes.  We are left with the challenge of reconciling these entities, that might  inform and adequately explain the pluralism in the "ontology" of mathematics. Is not this what L does in the first few chapter of Insight?

 

John

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[jaraymaker]

Hi,

 

in the light of the below discussion, I just added a comment on the already-out-there-now-real" etc, in our MS (see note 3) as follows 

 

Insight prepared Lonergan for what eventually became MiT. He wanted to develop his method based on the normal intellectual operations of theologians, and to make explicit how these conscious, intentional operations function when one does theology.[1] MiT seeks to implement the inherent impetus all humans have to be attentive, intelligent, reasonable, responsible and loving—and the myriad ways we stray from that drive. This helps ensure continuity in theology. The drive towards authenticity leads one to affirm reasoned positions and reverse false counter-positions,[2] as one searches for truth, in any intellectual endeavor. On the other hand,

Inasmuch as one accepts the counterpositions, one thinks of the real as a subdivision of the “already out there now real”, of objectivity as extroversion, and of knowing as taking a good look; similarly, on the counterpositions, the good is identified with objects of desire while the intelligible good of order and the rational good of value are regarded as so much ideological superstructure that can claim to be good only inasmuch as it furthers the attainment of objects of desire.[3]

The normative, universal nature of human intentionality that Lonergan explored led to his linking continuity with renewal in theology. For example, genuine continuity is achieved when theologians realize that people of different times and places operate under the same normative structure of their conscious intentional operations. One thus seeks for continuity not in concepts, but in the common cognitional operations possessed by people of different times and places. One discovers that intentional consciousness can build upon past achievements. While this implies development, it also means that contemporary work is linked to and continuous with the past. MiT fulfills Lonergan’s methodological ambitions by arguing that it is human minds who do theology. He intended his method to be a specific application of the general transcendental method (intentionality analysis) developed in Insight, which is why it applies to all who seek to understand something. Both Insight and MiT help readers discover the dynamic structure of their own conscious intentional operations. In both books, Lonergan assumes his readers are cognitive, moral and affective beings. It is crucial to realize that what Lonergan is after is, in fact, the method of all human attempts to know. His is not just another resource for solving particular problems, nor for addressing theological content.[4] By making explicit the conscious, intentional operations of theologians, Lonergan adds clarity, intelligibility[5] and precision to the theologian’s task. He does so by outlining what human authenticity is in relation to theological activity. MiT evaluates the degree to which a theologian operates in authentic human fashion. The word “operates” is crucial because Lonergan’s lifework was dedicated to clarifying how our minds operate. He defines method as a “normative pattern of recurrent and related operations yielding cumulative and progressive results.”[6] He adds that one has a method when there are distinct operations; each operation is related to the other; the relation forms a pattern which is verified as the correct way of performing the task; the pattern’s operations may be repeated indefinitely and the results of such repetition are cumulative and progressive.

Lonergan explains that explicitly using method in theology means conceiving it as “a set of related and recurrent operations cumulatively advancing towards an ideal goal.”[7]

---

[1] See Second Collection, 52, 268.

[2] Insight, 413. Lonergan develops “positions” from the experience of being “attentive, intelligent, rational and responsible”: 1) the real is the concrete universe of being; 2) the thinking person is known as the consequence of that affirmation of self as intelligent and self-critical, and 3) that objectivity is considered in that light and not as “a property of vital anticipation, extroversion, and satisfaction.” Counterpositions contradict one or more of these points in the positions.

[3] Insight, 647. This counter-position sounds like Milton Friedman’s insistence that a firm’s only responsibility to society is to maximize its profits. For Lonergan, the position implies experiential objectivity that is “the given as given. It is the field of materials about which one inquires, in which one finds the fulfilment of conditions for the unconditioned, to which cognitional process repeatedly returns to generate the series of inquiries and reflections that yield the contextual manifold of judgements.” Insight, 405-06. One might also ask where the 0’s and 1’s introduced by Boole as well as his concept of empty set as "foundation" and "complement to the universe," are not these profound insights of his not the "foundation" of the pragmatic digitalization now driving the world and making many humans ignore the true, deeper foundation on which Lonergan built his method? The empty set may be one form of the already already-out-there-now-real (Insight, 276) that neglects or downplays things, which exist as intelligible unity-identity-wholes. We are left with the challenge of reconciling these entities that might inform the notions of math as does Lonergan early in Insight.

[4] MiT, xii, 24, 254.

[5] In Insight, 101-02, in the section on the canon of relevance, Lonergan asks whether there ought not to introduced a technical term to denote “the intelligibility immanent in the immediate data of sense” that resides in the relations of things, not our senses, but to one another. He answers: “The trouble is that the appropriate technical term has long existed but also has long been misunderstood. For the intelligibility that is neither final nor material nor instrumental nor efficient causality is … formal causality. But when one speaks of formal causality, some people are bound to assume that one means something connected with formal logic; others are bound to assume that one means merely the heuristic notion of ‘the nature of …’, the ‘such as to …’, the ‘sort of thing that …’ If both these misinterpretations are excluded, what we have called the intelligibility immanent in sensible data and residing in the relations of things to one another . . . might be named a species of formal causality.”

[6] MiT, 4.

[7] MiT, 94, 125.  End quoting our MS     John

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To: loner...@googlegroups.com
Sent: 3/1/2022 6:45:26 PM Mitteleuropäische Zeit

.

[Doug Mounce]

John,

As my physics teacher used to say, "There are some books where I don't understand the first page, and there are others where I don't understand the first sentence!"  My stats teacher thought it was unfair for Boole to have his name forever attached to such a simple idea, but he took comfort in that he could never remember which Boole it was. 

As I understand it, Hamkin's main theorem is based on a formula of complexity where, in the language of set theory, ZFC proves a property of a finite set, and any transitive model of ZFC, and others, can be shown to be an empty set.  Now, if you understand that then congrats!  For a general dilettante foundation, I would recommend getting a handle on what ZFC is all-about, and you should want a somewhat particular, if necessarily vague, sense about what the Axiom of Choice might mean. 

Cantor's Paradise has a nice summary . . .

'The Axiom of Choice extends what we are comfortable doing with finite sets, is consistent with the other axioms, and makes a vast amount of mathematics work, and much of this mathematics is extremely useful.

"If you are looking for some kind of Platonist truth, then I think you are barking up the wrong tree. It is better to view mathematics as structure, much of which parallels the real world (such as Euclidean geometry giving real insights into objects which aren’t actually 2D and have perfectly smooth edges), rather than necessarily being some necessarily true statements.

"It works and underpins the mathematical objects we use to talk about probabilities, particle physics, and more."

https://www.cantorsparadise.com/what-is-the-axiom-of-choice-61347a0287c

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[David Bibby]

Dear Doug,

I’m not sure I quite understood that point (about the two directions in which research could have gone.)  What is undeniable is that the rise of set theory came to claim a central part in mathematics, reinforced no doubt by the reductionist tendencies of the time.  It seems almost inevitable, although once we understand that development, we may grasp it was not a unique possibility.

That would be an interesting exercise to express Lonergan’s concepts in set theoretic terms.  If set theory is so powerful that it can build practically anything from the empty set, then it should be possible.  The challenge to understanding insight, direct and inverse, in terms of set theory, might be to understand ourselves as sets first, and then understand insight as an intelligible operation within an intelligent operator.

Your approach to mathematics, beginning with formal systems, bears resemblance to Adrial’s definition (“expressions of systemic intelligibility via non-verbal signs”).  I think there is merit in that approach, but also limitations.  It seems to be an attempt to tame the complex, which has an excellent pedagogical motive, but we must not lose the sense of mystery at that which we do not understand.

Kind regards,

David

On 1 Mar 2022, at 00:21, Doug Mounce <doug....@gmail.com> wrote:

Hi David,

I think Hamkins did say that just-after set theory began to prove useful then research could have gone in two directions - either in empirical application to existing problems or in proliferation of speculative set theoretic possibilities, and researchers by and large chose the latter. 

I would guess that some concepts in Lonergan's work could be construed in set theoretic terms; the definition of an event, for example.

"What, then, is an event?"  Lonergan asks.  "The simplest answer is to say that it is a primitive notion too simple and obvious to be explained."  Of course, he goes-on to explain what is meant by, "the puzzling name `event'", and aggregate sets certainly play a role.

PS - if I can dovetail on your discussion of math then I'm inclined to begin with formal systems (i.e., what you can teach a computer - like arithmetic or chess) with semantic translation such that arithmetic informs us about numbers or geometry about space.  Logic is a form of math, in that definition, and Lonergan has the insight about how axioms must be continually added to shore up complete systems of logic.  FWIW

~/

https://staff.washington.edu/mounce/

On Sun, Feb 27, 2022 at 12:21 PM 'David Bibby' via Lonergan_L <loner...@googlegroups.com> wrote:

Dear Doug,

I agree, very interesting talk, thanks.  I quite liked the dots along the top which indicated the progress through the slides!

There was one slide which described the cumulative universe as a vast hierarchy of sets, sets of sets, sets of sets of sets … founded ultimately upon the empty set.  Contrast that with Lonergan’s explanation of living species in the development within genetic method:

“The fundamental element in emergent probability is the conditioned series of things and schemes; that series is realised cumulatively in accord with successive schedules of probabilities; but a species is not conceived as an accumulated aggregate of theoretically observable variations; on the contrary, it is an intelligible solution to a problem of living in a given environment, where the living is a higher systematisation of a controlled aggregation of aggregates of aggregates of aggregates, and the environment tends to be constituted more and more by other living things.”  (Insight, chapter 8, section 6, 2008/290)

I wonder whether we could form a set theoretic conception of that higher systematisation of aggregates of aggregates of aggregates…  The mathematical cumulative universe is built upon the empty set, but nothing concrete can be built from nothing.  The mathematical universe is abstract, ideal.  Within that abstract universe, there could be a pluralism of ontologies (as suggested by the speaker), but there was no mention of insight to pivot between the abstract and the concrete, to give it a sense of the real.  Knowledge of the real only arises when there is a complex, intelligent, species, able to make virtually unconditioned judgements by abstracting from the empirical residue of the universe in which it finds itself.  In other words, the epistemological question cannot be settled outside, but only within the set theoretic structure itself, when it describes a living, intelligent, autonomous, species, and genetic method is the best answer that species can give within the limitations of the coincidental aggregates of its experience.

Kind regards,

David

<Screen Shot 2022-02-27 at 19.41.22.png>

On 19 Feb 2022, at 01:18, Doug Mounce <doug....@gmail.com> wrote:

Interesting talk by Joel David Hamkins.  About two-thirds of the way through he discusses how the direction of research could have gone two ways after the power of set theory was uncovered.

http://jdh.hamkins.org/pluralism-in-the-ontology-of-mathematics-mamuphi-paris-february-2022/

Abstract: What is the nature of mathematical ontology—what does it mean to make existence assertions in mathematics? Is there an ideal mathematical realm, a mathematical universe, that those assertions are about? Perhaps there is more than one. Does every mathematical assertion ultimately have a definitive truth value? I shall lay out some of the back-and-forth in what is currently a vigorous debate taking place in the philosophy of set theory concerning pluralism in the set-theoretic foundations, concerning whether there is just one set-theoretic universe underlying our mathematical claims or whether there is a diversity of possible set-theoretic conceptions.

~/

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[David Bibby]

Dear John,

An example of building mathematics from the empty set would be the iterative conception S, set out by George Boolos (not Boole!) in 1989.

https://en.wikipedia.org/wiki/S_(set_theory)

“The Von Neumann universe implements the “iterative conception of set” by stratifying the universe of sets into a series of "stages", with the sets at a given stage being possible members of the sets formed at all higher stages. The notion of stage goes as follows. Each stage is assigned an ordinal number. The lowest stage, stage 0, consists of all entities having no members. We assume that the only entity at stage 0 is the empty set, although this stage would include any urelementswe would choose to admit. Stage n, n>0, consists of all possible sets formed from elements to be found in any stage whose number is less than n."

Another interesting notion within set theory is “the set of all sets”.  It is a paradoxical notion, because would it be a member of itself?  It is not a contradiction, but would involve an infinite recursion.  Compare that concept with Lonergan’s definition of the notion of being:

“Now our definition was that being is the objective of the pure desire to know.  Being, then, is (1) all that is known, and (2) all that remains to be known.  Again, since a complete increment of knowing occurs only in judgement, being is what is to be known by the totality of true judgements.  What, one may ask, is that totality?  It is the complete set of answers to the complete set of questions.” (Insight, chapter 12, section 1, 2008/374)

If everything is a set, then judgements are sets, and of those judgements, being is the totality of those that are correct.  The difference between being and the set of all sets is that the latter is completely abstract, the former is completely concrete.

Kind regards,

David

On 1 Mar 2022, at 17:45, 'jaraymaker' via Lonergan_L <loner...@googlegroups.com> wrote:

 

Doug, David,

 

I do have a question:  how can the whole of math be "ultimately" founded upon the empty set in the light of the below (the first half of which comes from Wikipedia as I googled "empty set")?

1) In math, the empty set is the unique set having no elements; its size (count of elements in a set) is zero. Some axiomatitc set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.

Any set other than the empty set is called non-empty.

In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). The empty set may also be called the void set. 

 

2) George Boole introduced the concept of empty set, or "nothing" as he called it, as the complement to the "universe" in his Mathematical Analysis of Logic (1847). His notations for them were somewhat boring, 0 and 1 respectively. Cantor wrote in 1880 "for the absence of points we choose the letter O".  End of quotes         John

 

In a message dated 3/1/2022 1:21:48 AM Mitteleuropäische Zeit, doug....@gmail.com writes:

Hi David,

I think Hamkins did say that just-after set theory began to prove useful then research could have gone in two directions - either in empirical application to existing problems or in proliferation of speculative set theoretic possibilities, and researchers by and large chose the latter. 

 

I would guess that some concepts in Lonergan's work could be construed in set theoretic terms; the definition of an event, for example.

 

"What, then, is an event?"  Lonergan asks.  "The simplest answer is to say that it is a primitive notion too simple and obvious to be explained."  Of course, he goes-on to explain what is meant by, "the puzzling name `event'", and aggregate sets certainly play a role.

 

PS - if I can dovetail on your discussion of math then I'm inclined to begin with formal systems (i.e., what you can teach a computer - like arithmetic or chess) with semantic translation such that arithmetic informs us about numbers or geometry about space.  Logic is a form of math, in that definition, and Lonergan has the insight about how axioms must be continually added to shore up complete systems of logic.  FWIW

 

~/

https://staff.washington.edu/mounce/

On Sun, Feb 27, 2022 at 12:21 PM 'David Bibby' via Lonergan_L <loner...@googlegroups.com> wrote:

Dear Doug,

 

I agree, very interesting talk, thanks.  I quite liked the dots along the top which indicated the progress through the slides!

 

There was one slide which described the cumulative universe as a vast hierarchy of sets, sets of sets, sets of sets of sets … founded ultimately upon the empty set.  Contrast that with Lonergan’s explanation of living species in the development within genetic method:

 

“The fundamental element in emergent probability is the conditioned series of things and schemes; that series is realised cumulatively in accord with successive schedules of probabilities; but a species is not conceived as an accumulated aggregate of theoretically observable variations; on the contrary, it is an intelligible solution to a problem of living in a given environment, where the living is a higher systematisation of a controlled aggregation of aggregates of aggregates of aggregates, and the environment tends to be constituted more and more by other living things.”  (Insight, chapter 8, section 6, 2008/290)

 

I wonder whether we could form a set theoretic conception of that higher systematisation of aggregates of aggregates of aggregates…  The mathematical cumulative universe is built upon the empty set, but nothing concrete can be built from nothing.  The mathematical universe is abstract, ideal.  Within that abstract universe, there could be a pluralism of ontologies (as suggested by the speaker), but there was no mention of insight to pivot between the abstract and the concrete, to give it a sense of the real.  Knowledge of the real only arises when there is a complex, intelligent, species, able to make virtually unconditioned judgements by abstracting from the empirical residue of the universe in which it finds itself.  In other words, the epistemological question cannot be settled outside, but only within the set theoretic structure itself, when it describes a living, intelligent, autonomous, species, and genetic method is the best answer that species can give within the limitations of the coincidental aggregates of its experience.

 

Kind regards,

 

David

 

 

 

 

 

<Screen Shot 2022-02-27 at 19.41.22.png>

On 19 Feb 2022, at 01:18, Doug Mounce <doug....@gmail.com> wrote:

Interesting talk by Joel David Hamkins.  About two-thirds of the way through he discusses how the direction of research could have gone two ways after the power of set theory was uncovered.

 

http://jdh.hamkins.org/pluralism-in-the-ontology-of-mathematics-mamuphi-paris-february-2022/

 

Abstract: What is the nature of mathematical ontology—what does it mean to make existence assertions in mathematics? Is there an ideal mathematical realm, a mathematical universe, that those assertions are about? Perhaps there is more than one. Does every mathematical assertion ultimately have a definitive truth value? I shall lay out some of the back-and-forth in what is currently a vigorous debate taking place in the philosophy of set theory concerning pluralism in the set-theoretic foundations, concerning whether there is just one set-theoretic universe underlying our mathematical claims or whether there is a diversity of possible set-theoretic conceptions.

 

~/

https://staff.washington.edu/mounce/

 

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[jaraymaker]

Please see this link  https://groups.google.com/g/lonergan_l/c/SdXNNLDTVlk  It leads to our discussions-efforts almost 5 years ago. Joel David Hamkins does seem to be a very versatile "thinker." ZFC is, of course, beyond my ken but the eight FS are part of L's solution to integrating knowledge while acknowledging mystery and such diabolical-totally-deluded actions of a mad Putin.

 

Frege is another would-be demystifying pioneer--if I understand the kern of this efforts correctly. See e. g. 

 

1) https://philosophynow.org/issues/106/Wittgenstein_Frege_and_The_Context_Principle

 

2) JOEL DAVID HAMKINS, " A deflationary account of Fregean Abstraction in Zermelo-Fraenkel Set Theory"

 

John

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From: loner...@googlegroups.com
To: loner...@googlegroups.com
Sent: 3/1/2022 9:43:23 PM Mitteleuropäische Zeit
Subject: Re: [lonergan_l] Pluralism in the Ontology of Mathematics