The Fax Machine in Your Skull

Estimated reading time at 200 wpm: 9 minutes

A question landed in a WhatsApp group the other day. Simple enough. What is one divided by zero?

One person answered at once. Undefined. That is the textbook answer. It is what a calculator, mathematician or the internet tells you. Clean, final, no fuss.

Someone else paused. They looked at the question a different way. What happens, they asked, if you divide by something small? Divide one by a half and you get two. By a tenth, a hundredth, a billionth- a zillionth – what next? Keep shrinking the denominator as close to zero as you like, the answer climbs without stopping. It runs off towards infinity.

Neither person was wrong. But the gap between them was not really about sums. It was about how each one holds an answer once they have it. One reached for the rule and stopped. The other watched the thing move and stayed curious.

Zero seems simple but it is not simply ‘nothing’. Infinity is not a thing that the human brain can hold in its mind except perhaps by symbols.

This is not a piece about maths. It is about belief. About how a lump of tissue behind your eyes turns signals into facts, and facts into certainties you would argue for. It starts with dividing by zero. It goes somewhere else.

The fax machine

Your brain is a fine piece of kit. It takes in light, sound, pressure and chemistry, and out of that raw traffic it builds a smooth, solid world. The world it builds feels immediate. It rarely occurs to anyone to doubt it.

But what you get is not the world. It is a copy of the world. A version. A model.

Think of a fax machine. The real document sits somewhere with its ink and its detail. The machine scans it, turns it into signals, sends them down the line, and prints a copy at the far end. The copy might be good enough to sign, or to act on. It is still a copy. And here is the part that matters. At the far end you usually never see the original. You work from the printout and trust it.

Your brain is that machine. The world is the document you cannot reach. The mind you call you – with what you know as your beliefs – is the printout.

There is one difference, and it cuts the wrong way. A fax machine does not mistake its copy for the real thing. It has no opinion. Your brain does mistake it. It builds its version of reality and then lives inside that version as if it were the reality. The copy is not flagged as a copy. It is taken for the thing itself.

That is not a fault. It is a necessity. If you felt the join between your senses and the world at every moment, you could not function. So the brain hides the seam. It works fast, spots patterns, keeps you moving. A rough map that gets you home beats a perfect one that arrives too late.

The cost comes later. In the rush of a normal day, the brain cannot tell the world from its picture of the world. It treats the picture as true. Most of what we call belief grows out of that one confusion.

Say I know and you usually mean my picture has settled. Say that is true and you often mean my picture matches the people around me. The real document stays out of reach.

The comfort of the quick answer

The person who said undefined was not wrong. By the rules of arithmetic, that is the answer. No number times zero gives you one. The operation is not allowed. The books agree.

Still, the speed tells you something. No pause. No second look. It was the answer that had been learned and stored, the one any examiner would accept. The safe answer.

There is nothing odd about that. The brain runs on shortcuts. Thinking costs energy, and energy is dear. Working everything out from scratch each time would wear you down by lunch. So when a question comes, the brain often reaches for the nearest stored answer and hands it over as fact.

This is why received wisdom travels so well. It is quick, it is easy, and it is rewarded. Agree with the room and you belong. Question the room and you do not.

Certainty is a comfort. A belief shared by others feels like safety. In time it stops being an idea and becomes part of who you are. Pull at it and you are not just testing a claim. You are risking your footing.

Why we guard the copy

Push on a firm belief and something pushes back. It might be a raised eyebrow or a raised voice, but it comes. This is not stupidity. It is self-protection. A belief that has grown into your identity is an anchor. Take the anchor away and you drift.

The defence is usually quiet. You keep company with people who agree. You notice what fits and skate past what does not. We call this confirmation bias, but it runs deeper than a mental habit. It is closer to a flinch.

I see the loud end of this in my own work. Sit with someone in the grip of a delusion and you meet conviction that no evidence can touch. Challenge it head on and you get anger, or a wall, or a belief that digs in harder.

But a delusion is not a different kind of thing. It is the everyday machinery turned up loud. The person who says undefined without a blink is not deluded. She is defended. The belief is comfortable. Authority taught it, school drilled it, friends share it. Why open a door you have no reason to open?

That is the human bind. The brain builds its version, treats it as real, and defends it. The same guarding that keeps you steady also keeps you shut.

So what is truth?

If the brain makes its own version of the world, and biology and habit and culture all shape that version, is truth left with any work to do?

I think it is, as long as we are honest about two things.

The world is real. It has weight and temperature and it does not care what you believe. Drop a stone and it falls whether or not you accept gravity. The ‘document’ exists regardless of what is printed in the mind.

And the copy is all you ever hold. You cannot climb out of your own head to lay the world beside your picture of it and check the match. You have the printout. That is the only page in the room.

So truth is not perfect agreement between copy and world. You could never confirm that. Truth is which way you are facing. It is movement towards copies that hold up better. Maps that get you there. Ideas that predict what happens next. Beliefs you can live by without falling on your face.

Here is the trap. A belief can spread widely and still be false. Take vaping. Belief that it is as harmful as smoking, or worse, has climbed to more than half of adults in Britain, up from around a quarter a decade ago. The evidence has not moved with it. It still points the other way. The belief grew through repetition and worry, not proof. Plenty of people now hold it. That has made it common. It has not made it true.

Medicine is full of the reverse, too. For years the settled view was that stomach ulcers came from stress and acid. Then two researchers found a bacterium, and one of them swallowed a dose of the bug to prove the point. H. pylori in most patients most cases studied was eradicated in about two weeks with bismuth and metronidazole, and if that failed he combined bismuth with amoxicillin. Ulcers that had been managed for years with acid suppression simply went once the bacterium was gone. The received wisdom was wrong that it was primarily about ‘acid’. It had been taught with confidence all the same. The world had not changed. The copy caught up with reality.

Back to zero

Return to the question. What is one divided by zero?

Undefined is right, inside the rules. It keeps arithmetic honest. A computer will hand you an error.

Infinity is also right, as a picture of the process. Shrink the divisor and the answer climbs and climbs. For a great deal of real work, that picture is the useful one.

Both hold. They answer different questions, and the trick is knowing which one you need and not muddling them. One gives you rigour. The other gives you something to build with.

The same goes for almost everything you believe. Some copies are sharp. Some are blurred. Some have been checked against other people’s copies and match. None of them is the original.

That is not cause for despair. It is a reason to hold your beliefs with a lighter grip. To wait a beat before you are certain. To leave room for a better copy tomorrow.

It is also a reason for patience with other people. If every mind runs on the same machine, then everyone is working from a printout, doing their best with the signals they got. The one who says undefined is not a fool. The one who says infinity is not reckless. They are two brains, reading two copies, in the only way brains can.

The point was never the sum. The point is the gap between the world and the version of it you carry about all day. That gap does not close. But you can know it is there. And knowing it is there changes how you hold everything else.