[SEQ RERUN] Math is Subjunctively Objective
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).
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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.
Quite often, one sheep plus one sheep equals three or more sheep.
Don’t know why this is being voted down. I smiled :)
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...
By this definition, both the continuum hypothesis and the negation of the continuum hypothesis are true in ZFC
This exposition would be much clearer if you reduced / expanded the concepts of “create correspondences between formal and real objects” and “ground a formal system in the territory”. Those look like they’re hiding important mental algorithms which the original post was trying to get at (Not the dot combining one. Maybe the one which attributes a common cause, a latent mathematical truth variable to explain the similar results of rocks and sheep gathering?). Do those phrases, “make correspondences” and “ground a system”, mean that we can stop talking about formal objects and instead talk about the behavior of physical circuits which compute all those formal things, like which strings are well formed, what the result of a grammatical transformation will be, and which truth values get mapped to formulas?
As it stands, I don’t see your point. You talk about a model which is true but doesn’t “say something” about reality. You don’t address whether things in reality “say something” about each other prior to humans showing up with their beliefs that reflect reality, i.e. whether there are things in the world that look like computations, things which have mutually informative behavior that isn’t a result of intermediary causal chains of physics-stuff jiggling each other.
Or maybe you did a little bit when you called sheep a map-level distinction? Physics clearly doesn’t act directly on sheep, but that doesn’t mean sheep can’t be a substrate for computing. Sheep are still there. It is a fact of reality that some fields contain hooved clumps of meat, even if we have to phrase that fact in terms of the response of visual-field segmenting and object-permanence-establishing neurons in the brain a person looking out upon the field.
I just wish I knew what you were getting at.
What feature of my mind makes me consider questions about the “meaning of math” meaningful?
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.)
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.
What he is really asking is “why do we think that Peano arithmetic is true?”.
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.)
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”.
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
Depends on your definition of true:
This statement is clearly not about accepting PA, but about counting sheep.
I wonder if intuitionism’s perspective sheds some light on this issue.
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.