The Difference Between Lying and Just Being Wrong

Once again I say, it is a strange world that we live in today.  My secondary monitor finally decided to give up the ghost.  It was very old and the primary had been replaced a few years ago so off to Best Buy to pick up a cheap, and I mean really cheap, small replacement monitor.  Well you can’t do that any more.  Oh you can buy a cheap monitor, but it won’t be small.  I guess nobody wants small monitors any more.  Anyway, my computer is really old and it uses a VGA port for the second monitor so I found an $80 HP VGA monitor at the second Best Buy and brought it home.

My video card recognized the monitor and my Nvidia software set it up properly and I installed it with the computer off, so when it fired up there was a picture on it.  Hooray.  But all of the Icons were scrambled because the new monitor was 16×9 and high resolution and the old one was not.  So the windows desktop was scrambled and it took a while to move all of the icons back to some semblance of order.  As I moved each icon, one at a time, it gave me a chance to review why I would have kept that folder, or shortcut and in many cases I deleted entire folders but one stood out and I had to look at what had been put there years ago.  And one of the videos in that folder is what has lead to this post.  So let me begin.

In 2014 I was particularly interested in Physics.  The reason is not important, but my interest in Physics led to an awful understanding.  Physics isn’t about what I thought it was about.  It is entirely about math.  Math.  I hate math.  And here I see all of the Physics professors at Nottingham University talking almost entirely about math and how they have managed to fit equations to all of the physical phenomena that they want to address in the videos that I wanted to watch.  But watch I did and finally I turned to the videos on Numberphile which is produced by the same guy at Nottingham University but for these videos he usually visits the math department.

Some of the videos are fun.  I’m not saying that I can follow them, just that they are interesting and to me what they point out is that the field of mathematics is fundamentally an abstract science (and yes I do think it is a science) that is about divining rules about numbers.  Basically, if you follow the rules that have been discovered by thousands of mathematicians over thousands of years, you will be able to use math.  You might, I most certainly won’t.  Remember I hate math.  And this brings us in a round about way to the conclusion that the infinite sum of all of the positive integers is … -1/12.

What.  That just can’t be right.  But here on the you boob is a thoroughly credentialed mathematician laying it all out in black and white.  He makes a series of contentions that seem to be correct, and then he applies that same logic to the final infinite series, the one in question, and he comes up with -1/12.  Just to be perfectly clear, as I know that the likelihood that anyone who reads this is going to be mathematically literate, what he is saying is;

1+2+3+4+5+6….for ever… = -1/12

I am pretty sure that each of you is scratching your head just as I did.  No matter how you think about it, the infinite sum of all of the integers is going to be positive, and it is going to be infinity.  It is not going to be -1/12.

So after 5 years and some six million views, one mathematician finally said enough is enough.  How could they get it so wrong?  Here is his video explaining where they went wrong and it really isn’t so complicated.

In the first part he explains why the thinking is wrong and in the second part he takes a deep dive into esoteric mathematics that you enter at your peril.  Basically, if you care, he explains that there are convergent infinite series and divergent infinite series and they are very different and the rules of what you can do with each are actually different.  The mistake that the Nottingham mathematicians made was in treating a divergent infinite series as if it were convergent.  This is where I would like to diverge from the arguments of mathematicians and enter into the world of philosophy.  This should be embarrassing.

The PhD’s in mathematics at Nottingham are not dummies.  They are really smart and they know a lot about mathematics.  It is their job to solve complex mathematical problems and to teach others how to be math wizards at Hogwarts.  And yet, what they believed to be true was wrong.  Not just wrong, but very very wrong on a very fundamental level.  Worse than that, they put out a video explaining what they thought to be true, and nobody who saw the video said “hey, wait a minute there.”  Not the guy making the video.  Not the other math profs.  Not the physics profs.  Nobody looked at this and waved their hand in the air…”question!”

But there is a process that is supposed to take care of things like this.  It is called the scientific method and it has been with us for quite a bit of time.

a method of procedure that has characterized natural science since the 17th century, consisting in systematic observation, measurement, and experiment, and the formulation, testing, and modification of hypotheses.

And there are steps that one must take to ensure that silly things are not introduced into the knowledge stream.  Think of these things as pollutants that ought not be allowed into the knowledge base that we all learn from.

When I pasted this list into this post I stopped and checked and yes indeed Step 7 is before step 1.  What!  Is this another Nottingham mistake?  No, it is not.  This list is saying that to follow the scientific method you not only have to do your own work, but you have to use the work of others that have come before you.  Each hypothesis is preceded by other hypotheses and each hypothesis is scrutinized like yours will be.  It is a circle that will be traversed every time a new idea is proposed; like the infinite sum of all of the positive integers is equal to -1/12.

So, to sum (pun intended) this up, did the mathematicians at Nottingham lie?  No, of course not.  They were not trying to be deceitful and they could not have benefited in the slightest by trying to deceive the viewers of that video.  They made a mistake, and it was one that they should have known better than to have made.  They were wrong.

If you finally make it to the end of the video that I have attached you will find some mind blowing mathematical results that relate directly to the concepts presented in the original video.  But not in the way that was originally intended.

I would also like to add that mathematicians have had to make some very bold additions to their tool set over the years.  This is because there are some things that are mathematical, but they are not numbers.  The most egregious of these is the concept of infinity.  Infinity is not a number and when a mathematician tries to put it into an equation, he may well manage to manipulate it using the rules of mathematics, but he is promoting a fraud.  And worse than that, he knows it and he is therefore, lying.  The sum of all positive integers is not positive infinity.  There is actually no answer to the question “what is the sum of all positive integers?”  This is because the infinite list of all positive integers has no limit.  There is no end to the list of numbers that one must sum so there can be no definitive answer to that question.  It has become a convention to assign the word infinity to this concept of things that never end, but it is not a number.

The smallest thing that can be identified as a specific thing is an atom.  Our science has now advanced to the point were images of atoms can be produced, and atoms in a matrix can be manipulated.  Yes, manipulated one by one.  We also know that these atoms are made up of things that are even smaller than atoms and we can produce these things in great quantities so that they may be sent around a ten mile long tunnel of super cooled magnetic fields and finally be smashed into each other, revealing even smaller things.  The number of electrons on an atom of hydrogen is one.  One tiny electron circling one tiny atom here on earth say.  But it isn’t the only atom of hydrogen.  There are many atoms of hydrogen here on earth alone.  And there are more still in the sun, which produces the energy that our earth functions on by crushing hydrogen atoms into helium atoms.

Now the sun is really big, and it obviously has an awful lot of hydrogen but it isn’t an infinite amount of hydrogen.  We can know this because there is even more here on earth and more in the atmosphere of Saturn, and beyond that, there are a lot of stars all of which are clearly burning their own hydrogen by converting it into helium.  But those stars are not infinite either as ours is simply one of a vast number of galaxies that we can see.  So all of the hydrogen, the smallest thing that we think of as a thing, in the galaxy, which is monumentally vast, and then finally in the universe.  That amount of hydrogen is not infinite.  But the number of positive integers, that is infinite.

So how can this be?  Well, the hydrogen is a real thing.  The stars are real things.  The galaxies and all that we can see, these are all real things.  The mathematics that are used to calculate things are not real things.  Originally numbers were created to represent real things.  I have one camel and you have one camel and if we walk together we will have two camels.  It was a method of counting.  Now it is a method of abstraction.  As mathematics moves further and further from it’s roots as a method of counting things, it moves deeper and deeper into abstraction.  It is not that this abstraction is not useful, just that it is not real.

And this is the fundamental difficulty that we find at the root of this video from Nottingham University.  The math and the manipulations that were needed were so convoluted that the result of -1/12 was not a surprise.  I am sure that lots of people were asked and every one of them said “yeah I guess it looks right to me.”  But they were wrong.

This is why there is a scientific method.  This is why people are encouraged to confront new ideas and to challenge those who would promote them.  The challenge is the thing that protects the knowledge base from pollution and it is a critical part of the process of learning how the world around us works.

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