Epistemological reductionism is, broadly speaking, an attempt to answer sceptical worries concerning epistemic "access". For example, how are we to have representational epistemic access to:

- states of affairs (e.g., future ones or long past ones),
- mathematicalia (e.g., infinite sets),
- moral properties (e.g., the property of being morally obliged somehow),
- possibilities (e.g., a possible world in which there are $\aleph_0$ members of the Beatles)
- the structure of space and time (e.g., the fine-grained topology of space below the Planck scale),
- causal connections (e.g., the connection between the magnetic field and force on a nearby electron),
- etc.?

*reduction*, such as:

1. If $p$, then it isEach of these reductionist proposals attempts to "close the gap" between the world and the mind. For example, if $p$, then rational inquiry would yield an epistemic warrant for $p$. This is the core assumption of Semantic Anti-Realism: that each truth isknowablethat $p$.

2. If a term $t$ has a value, then that valuecan be computed/constructed.

3. If a term $t$ has a value, then that valuehas been physically tokened.

4. If $P$ is a proof of $\phi$, then someone (or some community)grasps and accepts$P$.

*knowable*. (A similar view was advocated by Kant, Peirce and Dummett.)

However, Descartes, Hume, Russell and Popper all argued, in their own way, that these epistemic "gaps"

*cannot*be closed. (Descartes went on to try and close the gap by a complicated argument, set out in his

*Meditations*, involving God.) For the possibility of the obtaining of a state of affairs, of which we are

*non-cognizant*cannot, at least not with certainty, be ruled out.

That said, such a conclusion does not imply that one ought to be a sceptic. Human cognition, which I assume is neurophysiologically much like primate cognition (and in some respects like all animal cognition), presumably functions reasonably well in acquiring representational states which count as knowledge. Unfortunately, little is understood on this important topic in cognitive psychology, mainly because it is incredibly unclear what these representational states are. It merely says that we can't

*rule out*sceptical scenarios.

"2. If a term t has a value, then that value can be computed/constructed."

ReplyDeleteSomebody doesn't like non-constructive proofs. What does the word "value" mean in this context? Are we denying the Axiom of Choice here? One thing often ignored by AC-haters is that the consequences of denying AC are worse than the consequences of accepting it. If one is a constructivist, fine ... but your real line is full of holes and the intermediate value theorem is false. I can't live in a mathematical world like that and neither can the vast majority of working mathematicians.

If you want to have a continuum, then there must be an uncountable infinity of points that can never be defined, named, characterized, outputted by a Turing machine, approximated by an algorithm, etc.

I just don't understand the desire to name everything. Fact is there simply aren't enough names. Deal with it.

Thanks, Anon,

ReplyDeleteYes, I agree :) I'm thinking here of very low-level, computational terms in arithmetic, i.e., numeral terms in arithmetic, such as $(2 \cdot 3) + 5$, or $2^{2^{2^{2}}}$, etc. Ultra-finitists think that if a term $t$ has a value, then there should be an actual computation verifying it.

Yes, one could generalize the point to all sorts of valuations, and to cases where AC becomes relevant. For a case where AC isn't relevant, we could consider e.g., $\| GC \|_{\mathbb{N}}$, i.e., the truth value of Goldbach's Conjecture in the standard model $\mathbb{N}$. No one knows what this is. But normally we assume that each arithmetic statement has a truth value, even though we're not guaranteed to ever find out.

Cheers,

Jeff