Are quantifiers convenient fictions with fundamenta in re ? What does constructivism and dependent type theory have to say about this ? And functional interpretations ?
Universal quantification is either an abbreviation for a expression of finite conjunctive knowledge or else something about concepts and not pluralities or extensions.
Universal quantification is determined by a computable function.
A universal Turing machine or equivalent machine (we will not discuss finiteness arguments here) is enough to check any proof or run proof-searches. And all machines imply human intentionality.
The first principle of synthetic a priori knowledge: that a specific computation of one machine can be taken as showing a certain (non-finitarily verifiable) property of the computations of another machine (for instance, output X cannot be produced from any input). But this property itself may on the surface involve quantification in its expression. Again quantification needs to be seen as determined by a computable function or functional.
What is a typing judgment $t:T$ ? It is a statement that a given machine produces a certain output for input $t$ (inference) or $t$ and $T$ (checking).
Is there a particular philosophical interest in considering Boolean circuits or cellular automata (or neural network) models of universal Turing machines ?
The issue is difficult to describe simply but will try.
ReplyDeletewe have:
-analog circuits and discrete/digital circuits
-we have black_box and white_box
-concrete and abstract objects and activities
-Pavlov in neurophysiology distinguished between two types of systems/levels
-we have unconscious/subconscious and conscious actions
example of cooperation of two levels : discrete/conscious and analog/subconscious
How we want to play tennis - we learn based on conscious sequences of
movements - algorithms - whitebox.
However, the result of this learning is the formation of a subconscious analog mechanism
- neural network - involving motor coordination of the entire body.
A blackbox is created - with an unknown structure -
accessible to consciousness only as a whole - encapsulation.
The algorithm/consciousness here is the teacher for the neural network/subconsciousness.
When we want to impart a certain skill to someone we have to
convert the analog process into discrete/algorithmic
and in this form it can be made available to others for use.
Something similar to recording sound in CD format
- A discrete sequence of samples and an analog waveform
Analog activities are effective/necessary in the real - material - world
- in the artificial world of mind/technology discrete algorithms suffice.