# [SEQ RERUN] Math is Subjunctively Objective

2 12 July 2012 02:07AM

Today's post, Math is Subjunctively Objective was originally published on 25 July 2008. A summary (taken from the LW wiki):

It really does seem like "2+3=5" is true. Things get confusing if you ask what you mean when you say "2+3=5 is true". But because the simple rules of addition function so well to predict observations, it really does seem like it really must be true.

Discuss the post here (rather than in the comments to the original post).

This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was Can Counterfactuals Be True?, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.

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Comment author: 13 August 2012 03:22:37PM 3 points [-]

Why not call the set of all sets of actual objects with cardinality 3, "three", the set of all sets of physical objects with cardinality 2, "two", and the set of all sets of physical objects with cardinality 5, "five"? Then when I said that 2+3=5, all I would mean is that for any x in two and any y in three, the union of x and y is in five. If you allow sets of physical objects, and sets of sets of physical objects, into your ontology, then you got this; 2+3=5 no matter what anyone thinks, and two and three are real objects existing out there.

Comment author: 12 July 2012 03:53:36PM 1 point [-]

Quite often, one sheep plus one sheep equals three or more sheep.

Comment author: 12 July 2012 09:12:57PM -1 points [-]

Don't know why this is being voted down. I smiled :)

Comment author: 12 July 2012 07:47:11AM 1 point [-]

Two sheep plus three sheep equals five sheep. Two apples plus three apples equals five apples. Two Discrete ObjecTs plus three Discrete ObjecTs equals five Discrete ObjecTs.

Arithmetic is a formal system, consisting of a syntax and semantics. The formal syntax specifies which statements are grammatical: "2 + 3 = 5" is fine, while "2 3 5 + =" is meaningless. The formal semantics provides a mapping from grammatical statements to truth values: "2 + 3 = 5" is true, while "2 + 3 = 6" is false. This mapping relies on axioms; that is, when we say "statement X in formal system Y is true", we mean X is consistent with the axioms of Y.

Again, this is strictly formal, and has no inherent relationship to the world of physical objects. However, we can model the world of physical objects with arithmetic by creating a correspondence between the formal object "1" and any real-world object. Then, we can evaluate the predictive power of our model.

That is, we can take two sheep and three sheep. We can model these as "2" and "3" respectively; when we apply the formal rules of our model, we conclude that there are "5". Then we count up the sheep in the real world and find that there are five of them. Thus, we find that our arithmetic model has excellent predictive power. More colloquially, we find that our model is "true". But in order for our model to be "true" in the "predictive power" sense, the formal system (contained in the map) must be grounded in the territory. Without this grounding, sentences in the formal system could be "true" according to the formal semantics of that system, but they won't be "true" in the sense that they say something accurate about the territory.

Of course, the division of the world into discrete objects like sheep is part of the map rather than the territory...

Comment author: 12 July 2012 08:05:26PM 3 points [-]

when we say "statement X in formal system Y is true", we mean X is consistent with the axioms of Y.

By this definition, both the continuum hypothesis and the negation of the continuum hypothesis are true in ZFC

Comment author: 12 July 2012 02:54:20PM 1 point [-]

What feature of my mind makes me consider questions about the "meaning of math" meaningful?

Comment author: 16 July 2012 05:29:05PM 0 points [-]

People naturally tend to confuse the map and the territory, even when thinking directly about the territory. Thinking explicitly about the map requires a second-order map, and that just makes things even harder.

I would consider it a bug.

(Note that I am interpreting your phrase "questions about the 'meaning of math'" in a particular way, as "questions about the physical significance of specific mathematical facts and constructs". I chose this interpretation based on some of your previous comments, which suggested that you weren't yet totally on board with formalism. If, however, you instead meant "questions such as the one whose answer is 'mathematics is the map, physics is the territory'", then those questions are perfectly meaningful, and there is no reason your mind shouldn't consider them such.)

Comment author: 12 July 2012 05:43:42AM -2 points [-]

It's after reading posts like this that I'm happy to stick with instrumentalism, where questions like "is 2+3=5 true?" are meaningless. Peano arithmetic is a useful (meta-)model, and that's all there is to it.

Comment author: 12 July 2012 05:40:44AM 0 points [-]

Assuming Peano Arithmetic

By the definition of the S function and the + and = operators: S(S(0))+S(S(S(0)))=S(S(S(S(S(0))))) Further, by equivalence and the definition of 2,3, and 5: 2+3=5

If you do not grant Peano Arithmetic, then you need to provide alternate definitions for 2,+,3,=, and 5.

Comment author: 12 July 2012 05:51:28AM 1 point [-]

What he is really asking is "why do we think that Peano arithmetic is true?".

Comment author: 12 July 2012 10:11:51AM *  1 point [-]

I've never quite grokked this, is "true" an abbrevation for "true in this universe"? Because asking if a mathematical theory is "true" otherwise is just wrong.

(ETA: Any reason for the downvotes? This is a genuine question.)

Comment author: 12 July 2012 07:54:51PM *  1 point [-]

It seems to me that he's talking at least as much about the fact that S(S(0))+S(S(S(0)))=S(S(S(S(S(0))))) is a theorem of PA, and asking what it means for that to be "true".

Comment author: 16 July 2012 05:47:05PM 0 points [-]

Peano arithmetic does not have a truth value. Peano arithmetic provides the definition of 2,3,5,+,and =.

In other words, if you don't accept Peano arithmetic, then you cannot decode what I mean by 2+3=5

Comment author: 16 July 2012 05:54:55PM 1 point [-]

Peano arithmetic does not have a truth value.

Depends on your definition of true:

Because two sheep plus three sheep equals five sheep, and this appears to be true in every mountain and every island, every swamp and every plain and every forest.

This statement is clearly not about accepting PA, but about counting sheep.

Comment author: 12 July 2012 10:26:17AM -2 points [-]

I wonder if intuitionism's perspective sheds some light on this issue.