In the Introduction to Consequences of Pragmatism, Richard Rorty wrote of Platonists, Positivists, and Pragmatists.
In my philosophical interactions over the years, I see the contrasts as being Platonists, Conceptualists, and Synthetists.
There is synthetism as a movement in art [wikipedia:synthesiism], and there have been proposals for a Synthetic Philosophy [ wikipedia:herbert_spencer] [philosophynow], but philosophical synthetism comes from the perspective of the synthetic sciences: synthetic biology, synthetic materials, synthetic intelligence (which stands in contrast to artificial intelligence), computational matter. In Synthetism, the abstract (code) and the physical (matter) are synthetized.
Platonists believe that there is an external, etermal reality in addition to the material world. Mathematical Platonists, for example, believe in an immaterial, independent world of mathematical objects. Conceptualists hold “that universals exist only within the mind and have no external or substantial reality” [thefreedictionary:conceptualism]. But still there is a latency of Platonism in Conceptualists: They tend to think that logic and mathematics are “content-free” – that such subjects are independent of the material world and refer to immaterial things.
Synthetists (as materialists) do not believe in platonistic or mentalistic worlds, and that even logic itself is a synthetic enterprise, adapting to various domain-specific material contexts. [unconventiona_logics].
(The Materialist view of Conceptualism is it’s a bit like Schrödinger’s cat [S’s_cat]: Is Platonism alive or dead?)
What about infinite entities, like the “complete” set of natural numbers? For the Platonists, they “real”ly exist in an immaterial realm. For the Conceptualists, they exist, but in the “mind” (and are not material somehow – how could they be?). For the Synthetists they do not exist – unless they ironically do! (e.g. if there is the possibility for physical hypercomputers to exist.) The default for Synthetists is that infinite entities are Fictional entities [sep:fictionalism-math].
cf. “The Meaning of Pure Mathematics” (Jan Mycielski) [jstor:30227216].
Synthetists are prone to view mathematics as a combination of theory, computer-based proofs, and computer-based experiments.
Synthetists are aligned with materialists and pragmatists (“How did we ever get the notion of the mind as something distinct from the body? Why did this bad idea enter our culture?”
– Richard Rorty [rorty_2005].
Codicalism is Synthetism – with emphasis on the programatic aspect of the synthesis of code and matter.