Sharon's Three-Layer Solution to the Access Problem




Do you have an access problem?

If you're a classic platonist, people are always asking you about the access problem. ``We sure seem to know lots of things about mathematics. But if math is about objective and invisible abstract objects, how can such knowledge be explained? Don't you need to posit a spooky faculty of rational intuition to a benevolent creator who stocks our minds with true beliefs to explain such knowledge?''

But even if you're not a classic platonist, you might still have an access problem and not even know it!

Fictionalists face an access problem explaining:
  • How do we manage to come up with mathematical theories that are even consistent - and hence can characterize coherent fictional scenarios?
  • How come our intuitions about how to apply facts about what's true in the fiction (if 276+12=388 in the fiction that there are numbers, then when there are 276 apples and 12 oranges there are 388 fruit) to what must be true about the actual world, are so often right on the money?
 Plenetudinous platonists face these problems, and more:
  • How do we manage to come up with consistent mathematical theories?
  • How come our intuitions about how facts about mathematical objects relate to what's supposed to be true in the actual world are so often right on the money?
  • Boolos' example of consistent theories that are jointly inconsistent shows that it cant be the case that all consistent mathematical theories are true. So how do we manage to get the good kind of consistent theory?


Sharon's Three-Layer Solution Can Help

Whether you're a classic platonist, plenitudinous platonist, fictionalist (or even a structuralist or if thenist!), adding Sharon's Three-Layer Solution will banish your access problems, by giving you a nice naturalistic account of mathematical knowledge.

Layer 1: Causal interactions with the world secure practical success

Causal interactions with the world (Millian revision, and a nudge from mathematically shaped problems in nature) systematically explain how we got the kind of mathematical theories whose practical applications are a success - all without the unsightly psychologism and/or empiricism that characterized earlier naturalistic accounts of mathematical knowledge

Layer 2: The Metasematic Thesis

Layer 2: The Metasematic Thesis says that, in getting practically helpful mathematical theories it's not surprising that we managed to get ones that are largely correct, because most sufficiently practically helpful mathematical theories are substantially, if not entirely, correct.

This kit contains direct motivations for the Metasemantic Thesis, together with tools for attaching the metasemantic thesis to a wide range of different ontologies of mathematics, and a soothing balm for reference worries.

a) Direct Motivations for the Metasemantic Thesis
  • Analogy with Science: we don't find it surprising that in getting practically successful theories about e.g. bees we wind up saying many things that are actually true of bees.
  • More error-tolerance/less wholeism than science: Mathematicians mostly just care about the structure of a very limited number of facts and relations between mathematical objects, so it's more plausible to understand them as being correct about large swaths of mathematical fact, while being wrong about the nature of mathematical objects, than it would be to make the analogous move for natural scientists.
  • Wideness of the universe of mathematics: mathematical theories which posit structurally quite different kinds of mathematical objects, can intuitively all be true e.g. greek geometry, arabic algebra, modern set theory, modern category theory. Different ontologies of math can account for this fact in different ways (see below).
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    b) Tools to fit the Metasemantic Thesis into your preferred ontology of mathematics
    classic platonism: the Completeness theorem guarantees that there are enough mathematical objects in the universe of the sets to form models of all first-order consistent theories. fictionalism: there's a coherent fiction corresponding to every practically benign theory about what mathematical objects are supposed to be like, and how facts about them are supposed to correlate with facts about the empirical world plenitudinous platonism: saying that there enough are mathematical objects which all practically benign practices can be interpreted as talking about does not mean all consistent mathematical theories are true. For, there need not be a practically benign mathematical practice which would count as assenting to every logically consistent theory.
    c) A soothing balm for all but the toughest Kripkie-strength reference worries
    Notice that, by putting Layers 1 and 2 together we get:
    • a substantial body of correct beliefs and inference procedures about the relevant mathematical realm
    • modal reliability: in close possible worlds where you have a different mathematical practice, the naturalistic forces above ensure that you would still have a practically benign, and hence largely correct mathematical practice.

    Even Fodor admits this is enough to secure determinate meaning for our logical expressions, so it should be enough to secure a meaning for our mathematical expressions as well.

    Layer 3: A semi-deflationary theory of justification in mathematics

    A semi-deflationary theory of epistemic normativity explains how the reliable mathematical intuitions secured above can grant us knowledge. Layer 3 draws attention to a certain unprincipledness in the answer to epistemological questions like 'when is it OK to assume a necessary truth? why is OK to assume 2+2=4 if that feels obvious to you, but not the 4 color theorem?' The result is not only a common-sense-matching account of justification in mathematics, but a strategy for giving a satisfying answer to other kinds of skepticism (external worlds, other minds, the past, induction) as well.