Does 0.333333... equal 1/3 ?

lob wrote:

It's pretty interesting that people form strong enough platonist views about this sort of stuff to argue about it on the internet. I imagine most people replying on this thread thought about these sorts of questions for at most an hour or two when they first heard about real numbers in the 5th grade or whenever but seem to carry a kernel of their conclusions with them decades later.

It sounds like the people answering "no" are working with a definition along the lines of: ".333... is shorthand for the limiting process (i.e., algorithm), that starts with 0 at step 0 and then adds 10^{-n}*3 at step n."

People answering "yes" are using a definition like ".33... is shorthand for the limit (if it exists, and with respect to the real numbers) of the sequence (.3,.33,.333,...). The real numbers are are objects obeying such and such axioms, and according to these this limit must equal 1/3."

Ruxton Towers XC wrote:

lob wrote:

It's pretty interesting that people form strong enough platonist views about this sort of stuff to argue about it on the internet. I imagine most people replying on this thread thought about these sorts of questions for at most an hour or two when they first heard about real numbers in the 5th grade or whenever but seem to carry a kernel of their conclusions with them decades later.

It sounds like the people answering "no" are working with a definition along the lines of: ".333... is shorthand for the limiting process (i.e., algorithm), that starts with 0 at step 0 and then adds 10^{-n}*3 at step n."

People answering "yes" are using a definition like ".33... is shorthand for the limit (if it exists, and with respect to the real numbers) of the sequence (.3,.33,.333,...). The real numbers are are objects obeying such and such axioms, and according to these this limit must equal 1/3."

Yes! That is why I am upvoting all of you guys. I respect your passion, love of number theory, belief that arcane things are wonderful, and your commitment to downvoting me every time I say, "linguistically, it depends on how you define the concept of 'equal' in math vs. life."

You guys are thugging it out for five pages about whether or not "an unresolving sequence is equal to the fraction it is approaching?" or is the "fraction itself just shorthand for that sequence?"

If something can easily exist in life, physics, and geometry (such as 1/3), does that mean it must also be easily represented at a decimal?

It is sort of like saying "is pi equal to a number?" --- to my very first point, it depends how you define "equal" and "a number."

In other languages those definitions can shape how we think about math itself, so this debate reminds me of people arguing about "what it says in the Bible" when they are only reading the English translation of the Bible.

Math is a translation of some universal, platonic truths about the universe into human language. Geometry is one way of translating those truths. Algebra and calculus are two others. Normally those translations are all in agreement. But not always...

lob wrote:

bhah wrote:

Sorry, there’s no debate about 1/3 or pi. The topic of this thread is stupid trivial and you haven’t been on the right side of it. Math is very precise about its definitions and there are no contradictions. Linguistics is irrelevant.

Math isn’t necessarily truths about the universe, whatever that means, just stuff we made up.

Ruxton is very clearly debating the linguistics/philosophy of expressions like .3333... and not the mathematics of whether or not this expression equals 1/3 if it is a real number in a standard axiomatization. I think the same goes for a majority of the people who will answer "no" to these sorts of 1=.999... problems; they're not usually questioning what the axioms imply, they're questioning whether the axioms model reality and if treating expressions like this as numbers makes sense at all (whether they realize it or not, which they usually don't).

lob wrote:

bhah wrote:

Sorry, there’s no debate about 1/3 or pi. The topic of this thread is stupid trivial and you haven’t been on the right side of it. Math is very precise about its definitions and there are no contradictions. Linguistics is irrelevant.

Math isn’t necessarily truths about the universe, whatever that means, just stuff we made up.

Ruxton is very clearly debating the linguistics/philosophy of expressions like .3333... and not the mathematics of whether or not this expression equals 1/3 if it is a real number in a standard axiomatization. I think the same goes for a majority of the people who will answer "no" to these sorts of 1=.999... problems; they're not usually questioning what the axioms imply, they're questioning whether the axioms model reality and if treating expressions like this as numbers makes sense at all (whether they realize it or not, which they usually don't).

lob wrote:

bhah wrote:

Sorry, there’s no debate about 1/3 or pi. The topic of this thread is stupid trivial and you haven’t been on the right side of it. Math is very precise about its definitions and there are no contradictions. Linguistics is irrelevant.

Math isn’t necessarily truths about the universe, whatever that means, just stuff we made up.

Ruxton is very clearly debating the linguistics/philosophy of expressions like .3333... and not the mathematics of whether or not this expression equals 1/3 if it is a real number in a standard axiomatization. I think the same goes for a majority of the people who will answer "no" to these sorts of 1=.999... problems; they're not usually questioning what the axioms imply, they're questioning whether the axioms model reality and if treating expressions like this as numbers makes sense at all (whether they realize it or not, which they usually don't).

Ruxton Towers XC wrote:

lob wrote:

Ruxton is very clearly debating the linguistics/philosophy of expressions like .3333... and not the mathematics of whether or not this expression equals 1/3 if it is a real number in a standard axiomatization. I think the same goes for a majority of the people who will answer "no" to these sorts of 1=.999... problems; they're not usually questioning what the axioms imply, they're questioning whether the axioms model reality and if treating expressions like this as numbers makes sense at all (whether they realize it or not, which they usually don't).

Yes. This is my point exactly. That is why I brought up pi, for example. It isn't "a number" at all. It is a relationship between two things.

That relationship cannot be written down as a number, only as an term that expresses the relationship.

.33333... is not a number but an attempted decimal representation of a ratio (one third of something). It is not even really a "math problem" at all but rather a question of whether abstractions and symbols are the same as the things they represent. The word "cat" is not a cat, right?

I think it is obvious that I am very aware of how interesting and fun this topic is. So, yes, I "realize" where I am trying to take this conversation. :)

And I don't mind being on the "losing side" of this debate but the fact that it comes down to how we define things, does make it very interesting for logicians, philosophers, linguists, etc. and not just mathematicians.