# Building a Cabin: Dialogues on the Foundations of Mathematics

There is one concept that corrupts and deranges the others. I speak not of Evil, whose limited domain is Ethics; I refer to the Infinite.

-- Jorge Luis Borges

Mathematics, rightly viewed, possesses not only truth, but supreme beautyâ€”a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

-- Bertrand Russell

## Day One: Truth and Truth Makersâ€‹

Somewhere in the not-too-remote wilderness of an undisclosed mountain range, Skepticus and Bloviatius assist Mundanius in building his dream-cabin. The day is wearing on and fatigue sets in, but there are a few more things to get done.

Skepticus, and Bloviatius stand impatiently as Mundanius taps his head with a pencil. He pours over the sweat-stained hand drawn blueprints, perhaps only intelligible to him at this point.

*computerized*blueprints.

Mundanius furiously scribbles on his crumpled blueprints.

Bloviatius rolls his eyes and turns on his foot out of shear frustration.

*I*do the calculations then I

*know*they are right! How can I guarantee that your Auto-whatever program won't garble the numbers and have us waste time, money, lumber and hard work!

We just need to add one more bracing board here... Or was it the other side?

Skepticus gestures to Mundanius's crumpled set of blueprints.

Don't you suppose the same calculations are done behind the scenes in the software?

Appearing to not have been listening at all, Mundanius looks up from his blueprints to the place where the board needs to be placed to stabilize the scaffolded structure so their work can be finished for the day.

Begrudgingly, Skepticus and Blovatius follow his lead and began the necessary work to reach the day's stopping point.

A few moments later, they all kick back on the half-complete porch to watch the sunset and unwind after a hard day's work.

Bloviatius strokes his beard, looking at the stacked boards apparently in deep thought.

*tell us about reality*'.

*didn't*fit?

*correct about reality*? In other words,

*why is there a rule*that works that way at all?

Look, every true proposition must to have some thing that

*makes*it true, right? A truth-maker.

*why*is it true?

**Millennium Prize Problems**?

How can it be that its all rules, the rules are made by us, and yet we can't work them out.

*applicability*of mathematics. You can say it is a 'rule system' all you want, but that doesn't explain

*how*it allows me to successfully interface with the world!

Axiom one of "Schmathematics"...

I do actually disagree though. Mathematics is a

*special kind*of rule system because it has 'open variables' that apply to broad classes of things. For example, the Pythagorean theorem applies to ANY three 'things' that are lines and form a right-triangle. So, the boards in our example just 'stand-in' like the bishop 'stands-in' for the 'thing that is subject to the set of 'bishop-rules'.

How do you explain that when things 'stand-in' for an open variable in the rules of mathematics, they have applicability, but not when we substitute those same things into chess or Skepticus's "Schmathematics"?

There is another quandary for your account. How does just adhering to rules guarantee that your method generates truth? What if it starts generating falsehoods?

So, the one problem is that it doesn't guarantee 'applicability' and the other is that it doesn't guarantee 'truth'. I'm not satisfied with that.

I guess I would argue, regarding the 'standing-in' thing, that mathematics is just about the most general rule system we have, and so it makes sense that it's applicable because it is general.

How could we think 'wow these rules sure are telling us the truth' and also 'let's make up another one. Oh! It looks like that one is applicable too!' without acknowledging that it isn't our following them that makes them true or without acknowledging an astounding coincidence.

So, if the 'rules follow from rules follow from rules' and so on, then at some point our initial 'general rule(s)', as you'd have it, must have had some property that made the rules that followed from

*that*one

**true**and

**applicable**?

You act as if all of mathematics follows from some magical and minimal set of assumptions. I am more pluralistic than that.

Let's say I draw a circle in the dirt here.

Mundanius draws a circle in the dirt with a nearby stick

^{2}, right? Let's assume we never knew how to get the circumference of a circle or do algebra.

I am the first person to move some things around and show you C = 2Ï€ * r. I show you some graphical or algebraic proof and you believe me.

If you followed my proof exactly, by some rule, and it turned out that it was NOT 'applying' to circumferences generally, we just wouldn't play by that rule.

Bloviatius ponders the circle in the dirt for a moment as if looking more intently at it might solve the philosophical puzzle.

*even follow*those rules in the first place?

In your circle example, don't you need to follow rules in the first place to get to 'algebra' and thereby 'circumference'? Are you saying we are sort of 'empirically testing' those rules when we 'try them out' to be the first to find circumference?

*'hey guys, I found some rules about circles that are convincing and fit with everything else we know about them'*.

In general, I think it is a complex phenomena the 'emergence' of mathematics from human made rules-systems, but ultimately I think that is all it is. All of this excitement about deeper truths I think is just a confusion.

Almost standing up from his chair, Bloviatius looks up from the circle in excitement!

I think the only real mathematical objects are the ones we apply our empirical theories to. I guess I am still with Mundanius on the idea that we have ideas with 'open variables' that apply to general classes of things. To my knowledge, the only things that can go in there are physical things. What alternative are you suggesting?

Mundanius, you claim that rules make mathematical truths true and Skepticus, you claim mathematical truths have their truth-makers in empirical reality.

How can either of you you account for the existence of actual infinities in contrast with the existence of merely potential infinities?

Also, I am sure you both would agree that these truths are necessarily true and true in all possible worlds! How can that completely explain their truth? Even if you claim that some feature of the mind or some set of rules makes them obtain, surely there is a possible world in which our minds

*do not*have those features or where the rules were different? Then are the truths of mathematics false in that world!?

My position elegantly handles both of these difficulties that you seem to have overlooked!

**has**to exist. I could build three billion cabins, that doesn't mean that I am committed to the existence of two billion, nine hundred ninety-nine million, nine hundred ninety-nine thousand, nine hundred ninety-nine imaginary cabins.

If you use something in your theory or calculations you are ontologically committed to its existence. You claim there are infinities but deny they exist! Surely that is a contradictory position!

*as*a fictional character. Numbers or lines or whatever "exist", but that just means that we use those concepts.

We don't need some fancy schmancy "abstract reality" to explain that. I think you are just overcomplicating things here by being weirdly strict with words.

That still leaves the problems of universality and necessity though!

Mundanius casts a sidelong grumpy glance at Bloviatius who takes the signal to move his argumentative focus elsewhere.

The evening has gone on into twilight. In the momentary silence they all suddenly notice the as yet ignored hum of insects in the surrounding forest. Skepticus breaks the silence.

Suppose there is some ghostly realm of 'abstract objects' whose existence the propositions of mathematics depend on in some way. Suppose that that is what makes mathematical truths true. All of it.

Are these things physical?

Supposedly, knowledge comes from the senses or from reason! Pure reason alone could not guarantee our interaction with these mathematical objects or it would be the reason and not the objects that made them true!

So, it must be experience, but I fail to see how a 'non-physical' entity can be learned of from experience.

*directly*be acquainted with them but we can nevertheless have knowledge of them through the special faculty of intuition.

You can know that a "circle has only one side" without checking any circular objects, right?

And you would presumably agree that while no

*real circular thing*must be perfectly round, a circle is. What we refer to when we talk about circles are ideal! They cannot be 'circles in reality' to truly have the properties we claim they have. So some perfectly circular thing must be the referent of our concept of circles, which I argue is the abstract object 'circle'!

If you say they are merely 'collections of experiences' then that does not explain the existence of 'actual infinities' or explain why mathematical truths would be true 'in all possible worlds'!

Mundanius's head is slumped forward and he appears to have been asleep for several minutes. He lets out a sleep-addled mumble.

Skepticus carries on slightly quieter, but still intent on making his point.

Regarding universality and necessity, I can just claim that mathematics just

*is*contingent, but that is it just

*less*contingent than other stuff.

They carry on in heated whispers:

Claiming mathematics to be contingent is surely absurd though!

First, I wholeheartedly disagree with the 'contingency' of mathematics. That is unimaginable! Second, GÃ¶del's proof shows that mathematics is not reducible to logic for any system that can do basic arithmetic.

You may have the elegant solutions, but it seems only at the cost of a bloated ontology!

*non-physical*truths about reality!

You cannot explain as well how the truths of mathematics are true without a theory like mine! Mathematics would have no stable foundation!

Mundanius, now awaking, impatiently claps his hands to his knees and looks at both Skepticus and Bloviatius. It is time for bed. They all have grown weary working and arguing. No minds have been changed and no meta-mathematical truths have been undeniably asserted. There is only the foundation, for now, of a small cabin and future conversations.

Each of our philosophers seem to take a slightly different view on what makes mathematical truths true. Mundanius takes a deflationary account, Skepticus a hard-lined empiricist one, and Bloviatius the Platonist account. I myself tend to side with Mundanius, although I feel the pull of the other views on some issues.

Much more could be said on the foundations of mathematics, whether there is an epistemic need for one, the nature of the infinite, and what exactly mathematics is. For now, we will adjourn with our tired friends.