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I'm looking for recommendations for works that present roughly (or at least parts of) the following perspective on epistemology in mathematics. I hope having access to similar perspectives will allow me to be able to refine my own thoughts, and eventually present them in a coherent manner, interconnected with other works in the field and with correct vocabulary. I am a mathematical logician with no formal training in philosophy, so even telling me what the official names of various perspectives I'm describing would be helpful.

First off, when I talk of "knowledge" I'm interested in reasons why people assert things: is there a mathematical proof? is it the result of a scientific experiment? did they hear it from someone? is it from an official source? did they experience it directly? And how, given an assertion, it could be otherwise. This forms a nuanced version of "levels of knowledge".

As a formalist, within pure mathematics, I believe we "know" something (within a particular formal "term rewriting" system) if there is a sequence of steps following the symbolic manipulation rules in the formal system deriving that statement (or more practically, a less formal argument to convince us that such a derivation exists). As such, we have unparalleled levels of certainty: to question a pure mathematical result requires questioning the formal system itself or the concept of deduction (which is allowed, but it's telling that one must go this far to question a mathematical result). In exchange we forsake the ability to say anything about the real world: we can only make assertions about systems whose foundations we are certain of, i.e. only formal systems that we have created to have those particular foundations.

(The actual practice of mathematicians simply convincing each other that deductions exist adds an extra level of knowledge, where published mathematical results could be wrong because a mathematician made a mistake somewhere.)

Applied mathematics takes the other side of this trade: we get the ability to talk about the real world, but at the cost of dropping to a completely different level of certainty. A real-world situation is translated using a mathematical model to formal mathematics, where we can do some pure mathematical reasoning, and then a result is translated using the model to a real-world prediction. To question this result, one now also gets the ability to question the model being used. As far as we know, all mathematical models contain flaws or simplifications, so we lose a great deal of certainty in our results. The results of mathematical models could either be telling us things about the real world or about the limitations of our models.

I'm not particularly interested in a philosophical discussion or debate: I am already well aware of a wide variety of contrasting perspectives on mathematical philosophy. However, my research into formalism has mostly been turning up older works from the era of Hilbert or earlier. I imagine that with the modern practice of formalizing concepts for computers, there might be more sophisticated formalist perspectives. I'm hoping there is modern mathematical philosophy work that will help me refine and put into context my perspective, and would greatly appreciate any pointers in the right direction.

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  • This is music to my ears. I completely share your point of view about maths as a game with symbols which is certain, but doesn't say anything about the world.
    – Frank
    Jan 17, 2023 at 3:46

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Welcome TomKern,

I hereby attempt to give you the answers you are seeking. Don't hesitate to correct me if I misunderstand or am unclear. I'll give you the whizbang tour.

First off, when I talk of "knowledge" I'm interested in reasons why people assert things: is there a mathematical proof? is it the result of a scientific experiment? did they hear it from someone? is it from an official source? did they experience it directly? And how, given an assertion, it could be otherwise. This forms a nuanced version of "levels of knowledge".

Let's talk lay of the land first. Let's sort approaches in math philosophy with a broad brush in two camps borrowing a page from Dummett: there are philosophers who are primarily interested in mathematical realism, a position that traces back to Plato and his Forms: Frege, Russell, Hilbert, and Goedel are representatives. The other camp might be considered anti-realists: C.S. Pierce, Dedekind, Bennacerraf, Brouwer, and Hartry Field. The short version of it is that the 19th century German philosophers, as Dummett has noted in his book on the origins of analytical philosophers, are the heart of the debates involved, with Fregean antipsychologism being an important position to understand regarding the dichotomy between neo-Platonic, and alternative positions. There are those who still are proponents of logicism by way of maximizing reductionism of mathematics to logic despite the set back Kurt Goedel delivered with his incompleteness. Then, in the spirit of CS Pierce and more "Continental thinkers" there is a strong embrace examining mathematics as a languistic artifact and psychological special case. From what I can tell, mathematical realism is overwhelmingly popular with practicing mathematicians, and anti-realism is much more embraced by people with training in the sciences, though both sides admit some of the strengths of the other side.

So, be cautious regarding your curiosity about "knowledge". It's not wrong, but your implicit definition isn't philosophical enough among some analytic philosophers who are very eager to strip out the human agency in mathematical process. Continental philosophers seem less put out by it. Discussion of philosophical 'knowledge' generally centers around the technical discussion of justified, true belief, at least in analytic philosophy. Your interest centers around human motivations, which among analytical philosophers is construed more of a psychological or sociological question as a rule. I'm letting you know because contemporary philosophy of mathematics in some quarters is very analytical in the sense it is largely obsessed with the ontology of mathematics. Oystein Linnebo's Philosophy of Mathematics represents an example where review of topics is very much centered on set theory and the ontology of mathematical abstraction. My experience is that neo-platonic thinkers, in general, turn a blind eye to a number of dimensions of philosophy of mathematics. Hence, the preoccupation with such thinkers are such topics: Frege, Russell, Goedel, Hilbert; logicism; formalism; deductivism; abstraction; set theory and foundations; various systems of mathematics like calculus, algebra, etc.; axiomatic method; and structuralism. These are usually heavily involved and center around mathematical formalisms we've come to know and love: ZFC, NBG, proof and model theory, the suite of introductory formal logics, and tangentially computability theory and completeness. These topics lend themselves well to mathematical realism.

Anti-realism is a different set of topics: nominalism; fictionalism, structuralism, intuitionism, empiricism, and social constructivism as applied to mathematical practice are on the other side. These are often agent-centric in their approach, and have even shaped formal practices like defeasible, intuitionist, and constructivist logics. There's also views on mathematics from philosophers of science like van Fraassen. I'd say, given your post, I think you're interested in exploring the latter dimension, and you're getting frustrated coming up with resources on the first group.

I'll also note computer science's view towards mathematical practice isn't as oriented to understanding the origins of mathematics, since one of the primary tasks of the philosophy of computer science is to make sense of the distinctions and similarities between abstract computational models like the Turing Machine and the technical artifacts that are implemented by computer engineers and used in the physical world. While proof correctness advocates like Tony Hoare are very much formalists in their preoccupations and pursuits, and other camps of practitioners are very much interested in the discrepancy between design and implementation, there's a broad consensus that computer formalisms are models, and models are ideas that must be created in the physical world, and perhaps a widespread understanding that the mathematical formalisms of computer science are application.

Okay, so, I'm going to make a recommendation on a reading list.

First, arm yourself with a firm understanding of the naturalism of epistemology broadly and specifically (all references are SEP unless indicated otherwise):

These two articles do a good job of giving you philosophical traction to defend a psychologist (WP) position on mathematics.

Next, explore some prominent positions:

Then, some of the issues:

Lastly, if you want a great single-source, then I'd recommend Mary Leng's Mathematics & Reality (2010/2013)(GB) who takes the themes I've sussed out of the SEP and IEP above and weaves them into a single volume, and weighs in on the indispensibility argument herself in the appendix.

So, if you do this reading, I suspect you will not just refine your perspective, but you will leapfrog your own thinking, because the realist positions of mathematics that you are acquainted with, are IMNSHO, rather outmoded and indicative of an unfamiliarity with the broad developments of philosophy in the latter half of the 20th century.

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  • I've tried to restrain my passion for the topic, but if you have any additional questions, don't hesitate; I love this field of inquiry.
    – J D
    Jan 17, 2023 at 18:19

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