Visualizing Concepts in Math Logic [closed]

Question

If you were forced to speculate or offer anecdotal evidence, how would you say excellent practicioners of mathematical logic coneptually grasp statements like:

$$ \vdash ((P \rightarrow Q) \rightarrow Q) \rightarrow Q $$

...on an intuitive level?

Specifically, I'm curious if excellent practicioners will typically VISUALIZE such statements in any sort of way that doesn't involve mentally picturing the literal statement with its syntax as written above. For example, are Venn or Euler Diagrams typically a good way to go about things, or is that a bad idea in the long-run?

Personally, I know that with Analysis/Group Theory/Topology I have been successful in finding rough visualizations of pretty much every concept involved (with the understanding that mental pictures are not LITERAL representations of the relevant concepts and in fact can often be quite misleading if one is not careful with them); however, with logic I am finding this more difficult since the mathematical objects in question seem in many ways to be largely, explicitly syntactical. What this means is that the more I rougly convert formal statements into "intuitive" pictures, the more those intuitive pictures start to exactly resemble a specific interpretation which gets in the way of those sentences being explictly syntactical in nature.

As a consequence of this mess, I find myself getting confused in lacking basic intuitions over whether elementary statements are true or false before I attempt to prove them, unlike in other mathematical subjects.

In short, does anyone have any anecdotal or even speculative advice about how to be successful in visualizing (or NOT visualizing) the objects of Math Logic? How does one gain an intuitive feel for this subject?

Thanks!

Answer 1

I tend to think of logic in terms of algebraic semantics. That is, I typically visualise a Boolean algebra in which propositions reside, where implication corresponds to the order relation on the lattice, where conjunction and disjunction take two corners of a diamond and give you the extremeties and negation allows you to jump around in the lattice. Implication, when considered as a partial order, is a relation, hence can be visualised as a graph. Sometimes I think of chains being connected together. This applies especially in the case of equational reasoning. In fact, in computer science, one of the common data structures for implementing programs reasoning about equality is an equality graph.

The image above has a basis in the Lindenbaum-Tarski construction which starts with syntax and builds a Boolean algebra.

I would not follow the approach above when thinking about proofs. One has to be careful not to confuse truth and proof and I tend to think of proofs in more combinatorial terms. The approach above is also not what I recommend for thinking about specific models of a formula. But in those cases, the model probably has additional structure that aids in visualisation or intuitive reasoning.

Answer 2

Personally, I visualize that as a circuit with logic gates.

Of course my imagination doesn't bother with picturing the symbols, but both variables and operators are concrete nodes with connections, variables possessing "active/lit"(true) and "passive/dark"(false) state (plus "unknown/gray"(to be computed)), and operators possessing properties - inputs, outputs and methods of transforming them. I arrange them like a traditional circuit with left-to-right flow, inputs on the left, result on the right, then "assign a set of input values" and follow through.

Dissociating yourself from the concepts of "truth" and "falsehood" really helps There are two clearly distinct states, and operators analyzing and modifying them. There's no point mixing that with factual correctness of statements - it only muddles the issue.

Answer 3

Reify the math with concrete objects to go from passive learning to active learning.

Check this out, especially the Mathematics part. http://www.interaction-design.org/encyclopedia/aesthetic_computing.html