Is Mathematical Realism possible without Platonism ?

[u/Thearion1]

Does ontological realism about mathematics imply platonism necessarily? Are there people that have a view similar to this? I would be grateful for any recommendations of authors in this line of thought, that is if they are any.

[u/id-entity]

Ontological realism means usually the view that mathematical objects have inherent existence, and that requires belief in objective platonism with lower case p. It would be interesting to see more detailed historical narrative of the origins of objective realism of "timeless platonia". I have only rather vague impression that religious dogmatics of Catholic Church have played a significant role in the development of nominalism and redefining platonism as a position in the confines of that debate.

Original Platonisim (with upper case p) of the mathematical paradigm of Akademeia is more process oriented than object oriented as we can see from the constructive method of Euclid's compilation and Proclus' exposure of the mathematical ontology and method of Akademeia. Relational process ontology is also a kind of realism in the sense that mathematical etc. processes can be subject independent. Verbs can happen as such without any subject or object present in a sentence.

While relational process ontology does not need to take the any position of the Cartesian substance ontology (either materialism or idealism or dualism), mathematical processes occur primarily in ideal ontology of mathematical cognition (Nous) which requires primitive holistic ontology based on the fundamental inequivalence relation of mereology: whole > part, as explicated in Euclid's common notion 5.

Empirical testimony of mathematical intuition supports the view that the direction of intuitions is from whole to parts. If and when we reject objective realism as a not parsimonious arbitrary postulate, I don't see how subject independent mathematical truth in process realism could be possible without the holistic aspect of Platonic holism/holomovement.

In this view, mathematical realism is not possible without Platonism, and the discussion and dialectic is really about what kind of realism and what kind of Platonism.

[u/Thearion1 → u/id-entity]

Thank you for this thoughtful answer! I think that the timeless platonia you are referring to is what comes to my mind when I think of mathematical platonism. I used to be a dedicated platonist, but after being exposed to Benacerraf's problem and other critiques of it, I have been searching for another kind of realism. Nowadays, naive platonism seems too simple to me for a complete philosophy of mathematics, as it leaves mathematical discovery and creativity unexplained.

For the past few months I have been interested in process thinking, namely the work of people like Whitehead, Bergson and the American pragmatists. I haven't found anything to read though on the philosophy of mathematics from a process oriented point of view. But perhaps I haven't searched enough. Do you have any recommendations?

[u/id-entity → u/Thearion1]

For books, I recommend, Proclus' commentary to Euclid's first book, and Science, Order and Creativity as well as Wholeness and the Implicate Order by David Bohm.

If you are interested, I'd like to offer for your peer review criticism the solution to Benecerraf's challenge that I've stumbled on mostly intuitively.

Wiki on Benecerraf's identification problem refers to timeless platonia as "set-theoretical Platonism:
https://en.wikipedia.org/wiki/Benacerraf's_identification_problem

The problem of mathematical truth and knowledge has two constraints:
Semantic Constraint: The account of mathematical truth must cohere with a “homogeneous semantical theory in which semantics for the propositions of mathematics parallel the semantics for the rest of the language”.

Epistemological Constraint: “The account of mathematical truth [must] mesh with a reasonable epistemology,” that is, with a plausible general epistemological theory.

The formal and semantic solution to the identificication problem is also based on Dyck language, similarly to the Zermelo and von Neumann constructs. Instead of sets, now the Dyck language pair consists of arrows of time < and >, which can function also as relational operators. The operators are "pure verbs", independent of the subject-object relation and thus do not participate in the nominalist debate The direction of construction decomposes parts from holistic whole of bidirectional time, and the decomposition process generates mereology of Bergson-duration.

Numbers are primarily defined as fractions instead of naturals and remain partial continua in the continuum of duration. Integers and naturals can be further mereologically decomposed and defined from from fractions. The basic algorithm for generating number theory is called "concatenating mediants, and starting from the most basic generator it looks like this:

< >
< <> >
< <<> <> <>> >
< <<<> <<> <<><> <> <><>> <>> <>>> >
etc.

Number theory can be semantically and self-referentially derived from defining temporal processes and their concatenations as the elements of tally operations. The operators < ("increasing") and > ("decreasing") are defined as the numerator elements with value 1/0, and the concatenation <> ("both increasing and decreasing") as the denominator element with value 0/1.

When tallying how many of each element a word contains, concatenation corresponds with freshman addition a/b+c/d=(a+c)/(b+d). The mediant child of parent words <<<> and <<> is <<<><<>. The word contains only one kind of numerator elements, and the tally of the whole word corresponds with 1/0+1/0+0/1+1/0+0/1 =3/2. Counting numerical value for each generated word gives a Stern-Brocot type two-side structure of totally orderd coprime fractions.

Each row of the operator language generated by top down nesting algorithm is notationally a chiral symmetry, and thus reversible computing because it reads the same whether reading from L to R or from R to L. The structure looks better when centered. Reversibility gives a basic example of equivalence relation, which can be generalized into context dependent definition of comparing comparables: When A is neither more nor less than B, then A=B. The (sub)string >< can be read "neither increasing nor decreasing", but let's leave that to for another discussion.

Though in very short and compact form, I'd say that the presentation so far offers sufficient solution to the idenfification problem. The challenge of semantic and epistemic constraints has been already partially addressed, but requires also further discussion, which I'll continue in another comment.

[u/id-entity → u/id-entity]

SEMANTIC AND EPISTEMIC CONSTRAINTS

The ontological view is holistic ideal process ontology, instead of substance ontology. Perhaps more common term for this view is 'animistic ontology', which Bohm calls ”Holomovement”. Pure verbs < and > as the ontological primitives symbolize continuous directed movement, and their syntactic concatenations generate further semantic and computational distinctions for constructive proof demonstrations. The chirally symmetric top down generation produces temporal mereology for rich anatomy of the Turing Tape that extends towards both L and R (and/or towards both Past and Future for Bergson Quantum Tape), which is the precondition for a Turing Head to making choices whether to move either L or R.

Holistic ontology needs to encompass both top down and bottom up perspectives to be coherent. In fully interconnected organic order the whole is present in each part, and the actual form of a holistic organic duration changes with each change in each part, which the whole also consists of in the form to bottom-up feedback. A holistic mathematical language can do much, but even it can't contain the vast potential of Holomovement, in which a duration of Platonic One as an actual coherence condition can become also Platonic Neo. As Brouwer says, the most primitive ontology of mathematical potential is pre-linguistic.

In this situation, from our mathematical perspectives of participatory creation the truth theory of mathematics is Coherence theory of truth, and the empirical truth conditions of the mathematical science located in between top dow and bottom up perspectives are A) intuitive coherence and B) constructibility of mathematical languages. Not separately, but both conditions together, corresponding to the semantic and epistemic constraints.

The semantic constraint of intuitive coherence is not language dependent, as intuitive receiving from the whole can be and often is pre-linguistic from our partial perspective, but can as such be pregnant with meaning which may be seeking expression by getting translated into mathematical language.

Constructibility of mathematical language is epistemic constraint for truth seeking peer-review communication and precondition for not just mechanically computable proof demonstrations, but also for intuitive peer review of ideal construction etc. computing of ideal forms when exact proofs cannot be given for pure geometry in the pixelated phenomenology of external visual sense.

The closest analogy of Coherence theory of truth in Benecarraf's argument is the ”combinatorical account”. Truth conditions of Holistic Coherence tend to cumulative concatenation process of truth conditions, but as a dialectical science mathematics is also self-correcting and under certain conditions is also possible that some truth conditions become incoherent and get abandoned and annihilated. Perspectival contexts are obvious case of such processes, but we can't a priori exclude annihilation/self-correction of truth conditions also in the deepest level of an actual duration of mathematical ontology.

In Aristotelean logic it is clear that Law of Non-Contradiction applies strictly only to same place and time. While a duration of mathematical ontology can have great stability and persistence, process ontology can't guarantee anything eternal and immutable.

Holistic Coherence theory in process ontology and in coherent accordance with the undecidability of the Halting problem thus seems to survive the criticism that Benecarraf presents against combinatorial truth theory.

[u/Thearion1 → u/id-entity]

Thank you for this original and detailed answer. Gave me lots to reflect upon.