The discussion reminds of that story On being a bat (iirc) in Hofstadter/Dennets highly recommended The Mind’s I, on the impossibility of understanding at all what it is like to be something so different from us.
Thomas Nagel’s “What is it like to be a bat?” [PDF], indeed included in The Mind’s I.
I’m not sure quite what you mean by goals here. The most plausible interpretation I can offer is that:
Goals are the drives that cause behavior. “Because of goals X and Y” is an answer to “Why did you do Z?” (and not an answer to “Why should you do Z?”).
In this case, we ought to adopt the set of goals that (through the actions they cause) maximize our expected utility. Our utility function needn’t mention goal-achievement specifically; goals are just the way it gets implemented. Acquiring a goal uncorrelated with our utility function is bad, because value is fragile.
It’s not that the causes of the goal “get out of pain” are bad; it’s that the consequences might be. For a wide range of utility functions (most of which make no explicit mention of pain), a system that provided information about damage without otherwise altering the decision-making process would be more useful.
You’re right. PA is still consistent (i.e. has a model) even if
N = the set of strings of the form S*0
0 = the string "0"
S = the function that prepends "S" to its argument
fails to be one because of the way string concatenation works. There’s nothing mathematically special about theories that can use physical objects as a model.
(Minor quibble: the definition of addition isn’t an axiom. It’s just a relation definable in the first-order theory of arithmetic.)
If, whenever we took 2 bananas and stuck them together with 2 more bananas, we ended up with 3 bananas, 2+2=4 would still be ‘true’ in the abstract sense that it proceeds naturally from the axioms[.]
I’m not so sure of that. If putting 2 S’s next to 2 S’s got us 3 S’s, we could prove 2+2=3 in PA with the usual definition of addition:
(dfn) \a. 0 + a = a
(dfn) \ab. Sb + a = b + Sa
\a. SS0 + a = S0 + Sa = 0 + SSa = SSa
SS0 + SS0 = SSS0
Depending on the universe’s other rules for putting n things next to m things, we might also be able to derive “2+2=4”. In this case, we would decide that PA is inconsistent! Whatever the other rules are, this already shows that the “abstract” conclusions we can draw from a set of axioms depend on the way symbol manipulation works in our world.
I don’t think this is really a problem for your argument, but it’s an interesting complication. Many (most?) physical facts seem to have no influence on the symbolic manipulations we can use to derive them. For instance, symbolically computing a series for pi doesn’t seem to involve any actual circles the way shuffling symbols to add 2 and 2 in PA involves putting SS next to SS.
Graphviz is the LaTeX of graph-drawing tools. You’ll get professional-looking output immediately, but the customization options aren’t as discoverable as they would be in a visual editor.
If you plan on making lots of graphs or want them to look very pretty, I’d recommend it. If you’re just looking for a quick way to draw a graph or two explaining TDT vs. CDT it may not be worth the time relative to a generic (vector) drawing program.
(The Python bindings might make things marginally easier if you know Python and don’t want to learn more syntax.)