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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.

user21312
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    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 '23 at 12:48
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    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 '23 at 13:36
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    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. – Mauro ALLEGRANZA Nov 27 '23 at 14:03
  • @MauroALLEGRANZA My understanding is that "classical theory of concepts" is a framework for concepts i.e a theory about the human mind in some sense while Logical positivism is a theory of knowledge. While related they adressed different question as far as I understand. Accodring to a reference I read(https://philpapers.org/archive/LAUCAC-3.pdf) some problems with the classical theory are related to the analytic/synthetic division. But I am not an expert in philosofy or logic and so I might be going about this the wrong way. – user21312 Nov 27 '23 at 14:20
  • @Meanach The problem specification is rather sophisticated and elaborate. To clarify, the OP is looking for canonical language to conceptualize how mathematics conceptualizes. Concepts are taken to be units of thought, and the OP is looking for how mathematical concepts and the act of conceptualization in particular are believed to behave according to philosophers. – J D Nov 27 '23 at 16:33
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    @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 '23 at 16:42
  • @Conifold I realise that this field is a real swap of verions of models etc. According to my reference it is not "mandatory" to consider its structure to be of "containment type" see footnote in my reference page 8. Thank for the Carnap reference ill read about it! – user21312 Nov 28 '23 at 06:50
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    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 '23 at 13:07
  • @Conifold Ill give that some thought, thanks! – user21312 Nov 28 '23 at 16:51

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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.

J D
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  • 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 '23 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 '23 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 '23 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 '23 at 16:54
  • And if you haven't, understand Quine's argument in Two Dogmas where he undermines analyticity by rejecting reliable synonymy. That will save you from some grief when you get to the linguistics where linguists essentially reject that different syntax can in anyway lead to identical semantics. – J D Nov 27 '23 at 16:58
  • Thanks, this is gonna take some time to evalute! – user21312 Nov 27 '23 at 17:21
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    @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 '23 at 18:21
  • "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." Does this mean the mind arises only as the product of a natural process? But our models of natural processes arise as concepts in the mind! I find the origin of the mind is a mysterious Divine or Natural process. I call it the nameless God and/or Psychogenesis! Evaluation of truth-values in context, and the context in which concepts are tested, both arise as products of the nameless process. – SystemTheory Dec 27 '23 at 19:56
  • @SystemTheory It is as wonderous as anything that can be observed. If you believe it is the result of the Wonderous and Starchy Creator of All Things Pasta, then I wouldn't begrudge you that experience. https://www.spaghettimonster.org/join/ – J D Dec 27 '23 at 20:52
  • "Does this mean the mind arises only as the product of a natural process?" To answer this question, we would first have to resolve 'natural'. – J D Dec 27 '23 at 20:53
  • Psychogenesis transcends maps of belief. My conclusion, as an applied philosopher, and student of electrical engineering and intellectual property law, is that the concept of a natural source of cause only arises in contrast to the concept of a moral source of cause. Law maps sources of cause to moral or natural causes; to proximate or ultimate causes; and to Acts of God (supernatural ultimate cause). Philosophers only invent names for distinct patterns of expression in human discourse. Philosophy and science arise in the context of self-other communication which arises via psychogenesis. – SystemTheory Dec 27 '23 at 21:20