# Where is pi?

June 14, 2011

Psychotherapy

For example, take pi.

Pi cannot be found in nature – nothing can weigh pi grams, or be pi meters long, or last for pi seconds.  To see this, imagine that I told you that my lucky penny weighed pi grams, and that you owned a scale that went to infinity digits that could weigh my penny and allow you to disprove it. You wouldn’t have to turn on the scale to know that I was lying.  Because the universe cannot create something that weighs pi grams.  There are of course objects that weigh 3 grams, and that weight 3.1 grams, and that weigh 3.14 grams, and that weigh 3.141 grams – and so on and so forth.  But at a certain point, as one progresses down the infinity of numbers after that decimal point, one “runs out of bottom.”  Once you get down past adding one more atom, one more neutron, one more quark, one more Higgs boson (the carrier of mass, we think), one more string (do those things even have mass?) you are simply done. You have run out of bottom, because you have no more smallest conceivable unit of mass to add to my 3.14159…etc. gram object – in this case, I have claimed, a penny.  And so – long before you’ve reached infinity (math joke, sorry) – you’ve hit the wall. That penny of mine is not going to weigh pi, and furthermore, nothing else ever is.

You can play the same game with length and width and height, or energy, or time, or any other physical quantity.  And this is all assuming a classical world; forget that Heisenberg, where structure is probabilistic, means that you definitely couldn’t claim such a thing! It doesn’t matter what aspect of the physical universe you choose: in each case, you will find that pi cannot exist in the world.

And yet: pi “exists.”

I mean, we are talking about it right now, and understand it perfectly. Right?  We have, at least in principle, since at least elementary school when we learned to imagine taking spaghetti noodle circles and spaghetti noodle diameters and comparing their ratios, been able to think about pi easily for years. Which is to say, we can imagine pi.  It’s just not “out there.”  It’s “in here.”

Now we think we know where there is. But where in the world is here?  This points us squarely to the problem I have been hinting at with the last two posts – on statistics and zebras.  Non-empirical objects, often called rational objects, those famous unobservables and undetectables that make modern metaphysics so vexing: they don’t have spatial location. They don’t have mass.  They don’t have duration.  And so we can’t map them on a Cartesian axis or map of the universe, which means we don’t know where in the world they are.

Which leaves us with this vexing question. Since we know it must be somewhere, where is pi?

I am a psychiatrist (psychopharmacology and psychotherapy) specializing in the so-called "personality disorders," particularly narcissistic and borderline personality disorders. I was a Fellow and then an Assistant Professor of Clinical Psychiatry at Columbia from 2004- 2011, when I had an intellectual crisis-of-faith in which I stopped believing that neuroimaging could shed significant insight into the mystery of subjective experience. Since then I have focused on my clinical practice (at the Personality Studies Institute) and philosophical concerns.

### 23 Comments on “Where is pi?”

1. Justin Kiggins Says:

Well, it exists in our heads. Quite literally.

If pi truly is a distinct concept which humans can grasp AND such concepts have an underlying biophysical foundation in our brains, then it exists in the state of our brain at that moment when we are conceptualizing it.

Perhaps there is a pi-cell, like the grandmother cell or the Jennifer Aniston cell. (http://bit.ly/k2r66y)

Or perhaps there is no such dedicated cell, but it exists instead in a single state in the high-dimensional state-space of activity across the cortex.

Who knows what its manifestation is, but in that moment, when you conceptualize it, it exists in the physical world in your brain.

• Peter Freed, M.D. Says:

That mind exists in brain is just dogma. It’s easy to say when we point at the whole brain. But when you start to zoom in, where exactly is it? You say a cell. But the laws of physics are sufficient to explain the physical structure of the cell, and the laws of physics do not explain where the subjective experience of zebra is. So you have non-existent laws explaining an unobservable concept – in the head? How do you know? You have no grounds for saying this save convention!

• neuromusic Says:

certainly not “just dogma” or “just convention”

it is an axiom (which I stated in my solution to your “problem”), but it is an axiom which had been useful in developing interventions for human behavior, such as neuropharmocologies.

But you apparently have an axe to grind against empiricism, whereas I don’t see neuroscience as being particularly useful if we simply dismiss those phenomena which we have not observed as “the mind” which is somehow independent of “the brain”.

• David Says:

“Where’s the software in this computer? All I see is electronics.”

• Peter Freed, M.D. Says:

P.S. I’m not a neuroscientist!

2. Joseph Allgren Says:

Plato claims to know, not that I agree with him. Using pi makes the argument ore compelling, but isn’t the same really true for any number. Can we say with certainty that something is exactly 4 feet long or that it weighs exactly 6 pounds? Only up to the point when we say are measurements are fine enough and agree to ignore the fact that they are only approximations, no matter how finely determined.

• Peter Freed, M.D. Says:

Doesn’t Plato think numbers are in the world of form, not in the physical world?

3. Ellis M. Says:

Sorry, but there’s a fundamental problem with this post. Suppose you did own that perfect scale. Now, take an object that weighed exactly one gram and place it on the scale. Somewhere in the decimal places, the zeros would stop being zeros, for the exact same reason something could not weigh exactly pi grams. The universe deals in terms of fundamental units. The concept of a gram is just as arbitrary as pi grams. Neither can be perfect. Humanity could have just as easily designated a gram as being equal to what is now pi grams, and there would be no difference.
Also, continuing from your line of thought, if I can have a penny of mass one gram, can I not have a penny with a 1 cm diameter? Guess what? If I can have a penny whose diameter is exactly 1 cm, then its circumference is exactly pi centimeters. That’s the definition of pi.
Just because a number is irrational doesn’t mean it cannot be found in the natural world, at least, any less than any number can be found in the natural world. Numbers and measurements are human constructs.

Pi is the the ratio of any circle’s circumference to that circle’s diameter, so it exists in every meaningful way.

• Peter Freed, M.D. Says:

The ratio exists in your mind, not in the physical world. All that exists in the world is the circumference and the diameter. Not their ratio.

• Justin Kiggins Says:

As properties of the circle (which does not exist in the world), neither the circumference nor the diameter exist in the world. This either by your discretization argument here or by your zebra argument.

• Peter Freed, M.D. Says:

? – can you not see the circumference of a circle? Can you not trace it with your finger? If so, that line exists in three-dimensional space. That’s all I mean by “exists.”

• David Says:

Ratios are just as real of a thing as a length or a weight. There’s nothing in nature that says “3.1 grams” that we are reading off when we find a weight. We construct it using our conventions, and ratio in the same type of convention. This doesn’t change the reality of the thing.

I also don’t agree with your argument that because pi has infinite digits in our number system it’s impossible for something to be pi of something. In our number system, maybe the mass of a neutron or a proton for example has infinite digits, we don’t know. It’s possible for real things to have infinite non-repeating decimal places in reality.

As for the actual point of your article, pi is a “relationship”. Why do relationships have to exist somewhere physically to be real? You could make the same argument for any word or concept at all, say “dog”. In some views, a dog is a collection of a bunch of properties, a relationship. We don’t find if something is a dog (in this view) by comparing it to some ideal dog and testing how much it matches (just as we don’t find if something is pi by comparing it to something which has a time/length etc. of pi).

• Peter Freed, M.D. Says:

Wait, I don’t think that a neutron in classical physics can have infinite digits. Fermions end somewhere. I think in QM the mass cannot be pinned down, but the odds of infinite digits becomes vanishingly rare. We could post to physics forums and see what they say.

• mike Says:

Leaving behind the actual point of your article, your reasoning about not finding something that ways exactly pi grams is wrong. It’s unlikely that there’s a collection of objects that weighs exactly pi, but that has nothing to do with it having infinite digits. Any exact number has infinite digits, even if most of those digits are zero.

There’s no reason that the exact mass of some particle couldn’t be some small fraction of pi such that adding many of them together would give you a mass of pi. It would be no odder than having something with a mass that was a fraction of 1 gram such that adding a number of them together would give you exactly (out to an infinite number of trailing zeros) 1 gram.

• mike Says:

And a followup: as most numbers are irrational, it is highly likely that the mass of a neutron is in fact irrational and therefore has an infinite number of nonrepeating digits.

5. Greg Says:

Pi is not a thing; no one ever said it was a thing, nor even that it existed in itself. Pi is an expression of a relationship; that’s what a ratio is. (Note: I do not say that pi is the relationship itself.) You’re getting stuck in essentialist arguments, as though they were the only rebuttal to your positions – that is, you’re using them as straw men.

Let me put it this way: in the end it’s an imponderable, and really not a fruitful question, to ask whether numbers exist in nature or just as human constructs. If you take as your starting point the assumption that we are trapped in our perceptions, on what basis would you be able to establish a difference?

• Maritzka Says:

I started eantig a lot more fiber when I got pregnant, especially since I wanted to avoid pregnancy constipation. I usually eat it the form of wheat toast (the double-fiber kind) in the morning with breakfast but will down a glass of metamucil if I skip breakfast. Fruits and veggies are great too.BTW, the fiber one bars are awesome but don’t eat too many your first few days cause you’ll get nasty stomach aches since your body isn’t used to so much fiber.

• Vanderson Says:

to suffer from aaoistcinl personality disorder, a disorder most noted for the lack of empathy for others, and with some evidence of deficient functioning of the frontal cortex (e.g., reduced blood flow to the frontal cortex noted on brain imaging). John Wayne Gacy Jr. murdered over 30 young boys, and Ed Gein was thought to have only murdered two women, robbed numerous graves, and made clothing and furniture out of human flesh. One could easily argue that both had deficiencies in brain functioning/impulse control based on their actions.Individuals such as Herbert Weinstein raise questions for me about the reliability of using brain imaging, and brain damage (e.g., cyst) in defense cases. Weinstein only had one known violent/impulsive outburst, the murdering of his wife, and was later recognized as a “model prisoner”. Even though the PET scan had shown a frightening amount of the frontal cortex inactive, Weinstein wasn’t reported to have any difficulty in decision making, any other violent outbursts, or any deficits in the actions thought to be localized to the frontal cortex. I have met individuals whom were malingering in order to avoid legal consequences such as being jailed, and they can make very convincing cases.In the case of Tiegreen the evidence showed great support that his violent behavior likely resulted from his accident. It is likely though that no two brain injuries are alike. Since there is still much to be learned about the functioning of the brain, isn’t likely that what could appear to be similar brain damage in two different people may result in differing effects. This would then assume that brain imaging could be unreliable.

6. Steve Narwold Says:

pi exists as much as 3.1 exists. If you weight something that is 3.1 grams, it most likely is NOT actually 3.1 grams. Maybe it’s 3.100000000000000000000000001 grams! Numbers are not things at all. No number actually exists. Numbers are just a way to express quantity, and it is usually impossible to convert something in the real world to a number (especially in something like weight. If you go far enough, maybe the example above actually weighs 3.10000000000000000000000000100000873465 grams!)

My point is that if you can have 1/3 of a cup, then you can have pi cups (1/3 = 0.33333… to infinity, like pi = 3.141… to infinity). Pi is just like any other real number which, ultimately, is an idea, not a thing.

7. Sunny Says:

My favorite snack is stnrig cheese and a few raw almonds. I also like almond butter and a banana.I used to buy the single serving Sabra hummus packs when Costco carried them (for a very short time). The single servings were actually still quite big though if I remember correctly.