All of Daniel_Lewis's Comments + Replies

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.)

0anonym
I'm think you're exaggerating how difficult it is to use graphviz for simple things by comparing it to LaTeX. Consider this diagram in the gallery and look at how trivially simple the source file that generates that image is. I don't disagree that doing complex things can be difficult, but for graphs that consist of a handful of nodes and edges with assorted labels, and some boxes to group nodes together, it's hard to beat graphviz.

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 utilit... (read more)

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=... (read more)

2Nick_Tarleton
Nitpick but important: we couldn't actually prove it, just produce a convincing (in that world) false proof (that is actually a proof of a theorem in some other, inconsistent, system with slightly different inference rules).
2JGWeissman
We could figure out that our symbolic manipulation is inconsistent with the axioms based on the quirk you consider. There are more axioms needed to define Peano Arithmetic. Taking axioms 7 and 8 from Wikipedia, translated into your notation: \a. Sa != 0 \ab. Sa = Sb -> a = b (I also use the symmetry and transitivity of equality.) Note, from the axioms you stated: \a. S0 + a = 0 + Sa = Sa So, axiom 8 can be restated as: \ab. S0 + a = S0 + b -> a = b So, starting with your result: SSS0 = SS0 + SS0 = S0 + SSS0 But also, S0 + SS0 = SSS0 = S0 + SSS0 So, by the restatement of Axiom 8: SS0 = SSS0 And then using the original form of Axiom 8 twice: S0 = SS0 0 = S0 We have a contradiction of Axiom 7. Thus, it is proven that our symbol manipulation does not follow the Peano Axioms. This does not invalidate the Peano Axioms. It simply means that a given physical system does not follow them. Of course, it would be difficult for people living in an alternate universe where symbols really behaved this way to notice the distinction. And they likely would not have to, if all objects behaved that way; they would figure out some other math to represent their situation. And at some point, mathematicians in their ivory towers would develop this weird math that is really hard to write down and has no known application in the real world (both properties bringing great joy to these academicians).