3

Doing some work on theory of mathematical concepts and need a good framework that suits my own views. Is the classical theory of concepts, which seems to no to suffer very much when considered in relation to mathematical concepts, compatible with logical positivism's view on analyticity of mathematics?

If yes, does anyone have a reference?

I am using the following as my basic reference https://philpapers.org/archive/LAUCAC-3.pdf

There is a collection of critiques, some of which are not relevant for mathematics and some of which are related to the analytic/synthetic distinction. Hence by applying a epistemology in which mathematics is analytic I hope to be able to use the ideas of the classical concept theory as a foundation.

To elabtorate on my view on analyticity, which may be completely navie and wrong, but to me all true mathematical statement are "true by defintion"(a proof is manipulation of the objects as well as logic) and not because "facts about the world". This view might be compatible with the classical theory since this view concepts as "mental". Another things that makes me think the classical theory works nice with math is that mathematical concepts are well defined and we can thus consider a concept to be a collection of "features" stemming from any defintion of said concept.

9
  • 2
    Not really, mostly because the classical theory is too simplistic for mathematics. Mathematicians often do not assemble concepts from simpler concepts but define them "implicitly", by listing conditions and proving existence. So the definitional structure is more holistic than hierarchical and mixed up with inferential structure based on background axioms, which takes conceptual dependence beyond classical containment. These features are reflected in Carnap's Logical Syntax and other positivist works. Classical theory covers only Aristotle's syllogistic, not modern logic and mathematics.
    – Conifold
    Nov 27, 2023 at 12:48
  • 1
    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
    – Meanach
    Nov 27, 2023 at 13:36
  • 2
    The issue is quite complex and discussed at least since Socrates and Plato... How concepts are "made available" (known) to humans (human mind/language)? How concepts used by mathematics (described axiomatically or ...) are fruitfully applied to empirical facts? In view of these issues, Logical positivism's notion of analyticity does not add much. Nov 27, 2023 at 14:03
  • 1
    @JD So far I take concepts to be mental representations i.e "semantically evaluable mental objects" and according to the "classical theory" they are characterised by sets of necessary and sufficent conditions. The only problem that I dont think I will be able to get rid of is called "The Problem of Psychological Reality" in my reference. I seems impossible to argue or prove that concepts(even in math) actually work this way, I still think there is something called intution which is hard to explain in this way, at least formally
    – user21312
    Nov 27, 2023 at 16:42
  • 1
    I changed the link in your post to SEP article that talks specifically about Carnap's view of analyticity of mathematics, references there may be useful. The footnote says that containment can be replaced by "relations to defining features", but that still would not cover non-hierarchical dependencies induced by implicit definitions supported by axioms. One just cannot naturally arrange mathematical concepts into a pyramid with primitives at the bottom. Many are introduced in groups with "circular" dependencies among them, and it is those that make them meaningful, not reduction to primitives.
    – Conifold
    Nov 28, 2023 at 13:07

1 Answer 1

1

No, and logical positivism died because it lacks critical insights into how humans form concept systems. Post-positivist thinking is far more productive in this arena. I'll offer you one such system.

Since the source you offer in your post cites cognitive science, I'm going to offer the second-generation cognitive science of George Lakoff to characterize mathematical conceptualization. As a theory, it draws from Lakoff's views in cognitive semantics and is fully fleshed out in his Philosophy in the Flesh wherein he offers the full scope of his and colleagues' position that he labels embodied realism which is a flavor of embodied cognition. His work on conceptual metaphors is quite extensive, however, he ventures into mathematics in his and Nuñez's Where Mathematics Comes From (WMCF).

I won't be able do justice to the theory here, but it starts with the idea that fundamental concepts are formed in discrete units of neurological computation. If you're familiar with computational neuroscience, you'll know there has been many excellent efforts to characterize how small neural networks can be modeled. Neurological computations as a basis of language has the benefit of aligning with NCCs. I would argue at this point in cognitive science, to see the mind as a product of anything other neural and chemical activity that captures the essence of the changes of state in the body is naive.

WMCF doesn't get into the nitty gritty of neural encoding or calculus-based description of dynamical systems. It waves it's hand in the air just pays tribute philosophically. What it does do is start with the idea that mathematics is essentially an aspect of experience that traces its formal semantics to four fundamental neural capacities, called the Fundamental Conceptual Metaphors. For instance, the Metaphor of Containment is the claim that our ability to see experience space as volumetric is what underlies the notion of a set. The Metaphor of Infinity is very much similar to the mathematical constructivist notion that infinity is a potential infinity generated by iteration. Our ability to track motion in our environment underlies the notion of the infinitesimal and the epsilon-delta definition because the latter is an infinite point-wise description of continuous motion along a plane. And so on.

11
  • You said: "So far I take concepts to be mental representations i.e "semantically evaluable mental objects" and according to the "classical theory" they are characterised by sets of necessary and sufficent conditions. The only problem that I dont think I will be able to get rid of is called "The Problem of Psychological Reality" in my reference. I seems impossible to argue or prove that concepts(even in math) actually work this way, I still think there is something called intution which is hard to explain in this way, at least formally"
    – J D
    Nov 27, 2023 at 16:49
  • Yes, the classical theory uses real definitions based on N&S. Lakoff follows Wittegenstein and Rosch down an alternative path of prototype theory. As far as intuition, of course, what one has to do is reject anything that leans towards the language of thought paradigm of intuition. Connectionist models like those of ML are better formal semantics for understanding what happens under the hood. Psychological reality is no problem at all, because reality can be dealt with as a domain of discourse that applies to self-awareness that bridges...
    – J D
    Nov 27, 2023 at 16:52
  • the gap between empirical experience and the language that describes it, mathematics. In fact, there are actually a plurality of languages that describe that experience as made manifest by Curry-Howard-Lambek.
    – J D
    Nov 27, 2023 at 16:52
  • The best book I've found so far to see the representational theory of mind from a connectionist perspective is Shea's Representation in Cognitive Science. If any of this makes sense with you, or you want to escalate the conversation, let me know. I'm currently involved in NLP work regarding semantic systems, natural language ontology, and categories, so I'm knee deep in resources in this direction.
    – J D
    Nov 27, 2023 at 16:54
  • 1
    @user21312 When you're done with that introductory material, I have some other suggestions for you depending on how the theory integrates into your personal metaphysics. There are topics like categorial grammars, quantifier variance, and the Sellarsian "categorial given" that you might find relevant. ; ) It's a never ending rabbit hole of papers and theories if I'm honest.
    – J D
    Nov 27, 2023 at 18:21

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .