Throughout this blog, I’ll refer many times to getting my answers through inductive reasoning, and it will probably benefit you to know what that is. Inductive Reasoning is, along with deductive reasoning, one of the two methods of figuring things out with logic. While deduction makes a more airtight case than induction, it is still an extremely useful tool that would be foolish to ignore. It is frequently used in mathematics, and may perhaps be necessary for any problem involving infinite numbers.

A common example of inductive reasoning is used in figuring out whether the sun will rise tomorrow. It rose this morning, and yesterday morning, and every morning of my life, and every recorded instance of mornings in history. Because the pattern fits so many intervals uninterrupted and without a single pattern break, it is said to be a *strong* inductive argument.

Mathematicians also use inductive reasoning to figure out, for example, what you get when you divide 1 by 3. We all know it’s (roughly) .333… , which is read as “point 3 repeating”. How does one reach this conclusion? We can’t keep dividing 10 by 3, dropping the remainder of 1, adding a 0, and dividing again by 3 over and over again. There isn’t enough time to do the whole problem, and it would be pointless to do it infinitely anyway because we all spot the pattern rather quickly. Because we know that this pattern will repeat every time without fail, we inductively reason that, no matter how many digits we add to the problem, we’ll get nothing but threes.

It’s important to note that this pattern doesn’t have a reason to break or interrupt — what is true about it at the beginning is true at the end, even if we keep it up into infinity. Strangely, some mathematicians would prefer to believe that the pattern does suddenly change when a number becomes infinitely long. Why? We’ll explore that in further blog posts, but the important thing to note here is that inductive reasoning will tell us what a pattern will do for any length of math problem, even an infinitely long one, and they are all *strong* arguments because there are an infinite number of examples in which each one works.

One final note: While I welcome criticism, please do not claim that I am wrong unless you can explain how I could have been right. If you don’t believe that inductive reasoning is useful for figuring out a problem like 1/3, please tell me how one does figure out such a problem. If you don’t have an alternative, there’s a good chance that there isn’t one.

Unfortunately, I’m afraid that your conceptualization of mathematical induction is rather flawed.

You have described the kind of induction used in science, where we draw conclusions from observations.

Induction in MATHEMATICS is a very well-defined technique. We prove a theorem true for a base case x, and then we show that if the theorem is true for any x, then it is also true for any y > x. (There are some variations, but that’s the gist of it.) Then the theorem proves itself infinitely many times.

Mathematics operates according to the rules that people define. You can’t just add your own rules to elementary algebra when they aren’t consistent, because then all kinds of other things break down. Sometimes we get unintuitive results; accept it. Or invent your own algebra.

Thanks, Noah. It’s a great reply. I would argue that I’ve shown how scientific induction is also used in math (to get answers like .333… when dividing 1 by 3), but you still may be right in that both techniques are used similarly in places and differently in others.

I can totally accept that infinity may be an exception to the normal rule, and perhaps it isn’t just a case of special pleading. But I’ve specifically used examples in other posts of cases where mathematicians don’t have problems with these same rules. For instance, no mathematician has said “you can’t add 1 to .999…” just because there’s no last digit to start the math with (a common argument against my methods), and when I demonstrate the techniques that I assume are used for doing such math (starting from the left, scientific induction) then I’m told that it doesn’t work when I use the exact same method to prove something that isn’t universally accepted (such as subtracting 1 from .999… in “a mind-blowing proof”). Before coming to the conclusions that I did, I supposed that infinity could just be a special case, but I see so many examples where there’s inconsistency in that reasoning.

I don’t believe that I am “creating my own algebra”. I think that the method I’ve employed here is simply more consistent than what mathematicians are currently using when developing models of understanding infinite numbers.

Upon further reflection, I noticed that your definition of “induction in MATHEMATICS” is actually the definition of other arm of logic, deduction. While deduction is useful and certainly plays a role in any logical endeavor (including math), that doesn’t make induction any less a part of drawing logical conclusions.

Also, I said that I’m not trying to “add my own rules”, but in a way I am. Most of what I describe is not a tale of what mathematicians are doing but rather what they

oughtto be doing, not by my personal opinion but rather as an extension of what they are doing with finite numbers. I think Cantor’s methods misled many mathematicians, which is somewhat ironic since the general consensus and popular belief are often cited as evidence against my claims, whereas they could have been cited for my claims before Cantor revolutionized the topic.http://tvtropes.org/pmwiki/posts.php?discussion=13071629540A12080000&page=14

I started a discussion about your ideas on the tvtropes forum.

Thanks. Any publicity is good publicity, I suppose.

I don’t know about the general population but mathematicians have deductive arguments for why the decimal expansion of 1/3 is .3333….

1) The first digit will be a 3 because it is found by dividing 10 by 3

2) If a digit d(n) is known d(n+1) will be the same number because it is found using the same method

c) All digits in the decimal expansion will be a 3

It’s a fine logical argument, and I don’t disagree one bit with your conclusion. However, it’s almost identical to the logical argument I made for the remainder (except I cleaned up the grammar a bit):

1) The remainder after finding the first digit is a 1.

2)The remainder for d(n) is known by taking d(d/10+1) and will continue to be found using the same method.

3) All remainders will be a decimal expansion of 0’s followed by a 1.

If you believe that your argument is true (as I assume we both do) then you also have to accept mine as true. Keep in mind that I’m not arguing that the answer is anything different than .333…. — you get nothing but 3’s, even when taken to infinity. However, it still carries a remainder that never becomes 0, even when taken to infinity.

You said mathematicians use inductive reasoning to find the decimal expansion of 1/3 and my point was they can use deductive reasoning to do it too.

The mathematical induction I was using only works for natural numbers so I can only say things are true for finite natural numbers.Wikipedia says there’s a form of transfinite mathematical induction using the ordinal numbers but I’m not very familiar with it.