Talons Philosophy

An Open Online Highschool Philosophy Course


The Penrose Tile, Continued

Zonohedron #3: Penrose tiling

Image courtesy of Flickr user Andrew Hudson.

I took Vincent’s comment on my recent post about the Penrose Tile to some mathy friends of mine online, asking what they thought of his extension of the concept:

I feel that the golden number does appear in several ratios, but at the same time I feel that we are searching for that ratio in nature, rather than trying to disprove it. It feels like were saying look at all these examples that follow the golden ration (I totally support the golden ratio ideology, its just that it seems too convenient) and we aren’t pointing out any that disprove it. In science, empiricism, logic, rationalism, and basically everything that creates theories; the goal is to disprove your theory and therefore prove it.

I don’t know of any examples of this theory appearing false of course, but I don’t know if we have searched for any. Therefore, I wonder, is there anything that disproves this golden number theory, the magical sequence of 0,1,1,2,3,5,8,13,21 . . .?

David Wees is a math teacher who used to be based in Vancouver before relocating to New York, and had this to say as a reply to Vincent’s thoughts:

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Rationalism, the Paradigm Shift, and the Penrose Tile

KITES & DARTS: British mathematician Roger Penrose created a plane of beautiful, endless variation with just two shapes, kites and darts, seen here in blue lines. Image by Dominique Fung

I came across this article in Nautilus that seems to tread some familiar territory in terms of a few of our discussions during metaphysics in the last while.

Given that Fibonacci seems to appear everywhere in nature—from pineapples to rabbit populations—it was all the more odd that the ratio was fundamental to a tiling system that appeared to have nothing to do with the physical world. Penrose had created a mathematical novelty, something intriguing precisely because it didn’t seem to work the way nature does. It was as if he wrote a work of fiction about a new animal species, only to have a zoologist discover that very species living on Earth. In fact, Penrose tiles bridged the golden ratio, the math we invent, and the math in the world around us.

However, an added piece of this idea relates directly to a concept at the heart of our epistemology unit, as relates to the idea of a paradigm shift:

It was as though Penrose’s fanciful mathematics had forced itself into the natural world. “For 80 years, a crystal was defined as ‘ordered and periodic,’ because all crystals studied from 1912 on were periodic,” Shechtman says. “It wasn’t until 1992 that the International Union of Crystallography established a committee to redefine ‘crystal.’ That new definition is a paradigm shift for crystallography.”

It was more than mere mental inertia that made it so hard to understand and absorb Shechtman’s discovery. Aperiodic crystalline structures weren’t just unfamiliar; they were supposed to be unnatural. Remember that the placement of one Penrose tile can affect things thousands of tiles away—local constraints create global constraints. But if a crystal forms atom by atom, there should be no natural law that would allow for the kind of restrictions inherent to Penrose tiles.

As we continue to argue the merits of empiricism versus rationalism, doesn’t the example of the Penrose tile present a case of rationalism leading the way?



Mathematical Platonism: Have Some Delicious Pi

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices.

Øystein Linnebo, Platonism in the Philosophy of Mathematics

In my previous blog post, I wrote at length about what numbers are (and consequently, whether my math homework exists or not). Unsurprisingly, I was not the first philosopher to ponder the properties of mathematical objects. The inspiration for the theory of mathematical Platonism dates back to Plato and his Theory of Forms, as one can infer from the name of the theory.

However, mathematical Platonism is not directly derived from Plato’s Theory of Forms; instead, many of its principles are based upon the work of the 19th-20th century mathematician and philosopher Gottlob Frege. Frege’s works have been adapted and intertwined with similar ideas, and an modern expert in the field of mathematical Platonism is Øystein Linnebo, whose ideas I will be quoting at length in this post.

plato world

Image taken from abyss.uoregon.edu and used/modified under Creative Commons License.

Øystein Linnebo is the author of Platonism in the Philosophy of Mathematics, and he begins by describing three core attributes of the theory:

Mathematical platonism can be defined as the conjunction of the following three theses:

There are mathematical objects.

Mathematical objects are abstract.

Mathematical objects are independent of intelligent agents and their language, thought, and practices.


A decent grasp of these three ideas is essential for an understanding of mathematical Platonism, so I’ll go through them one by one in more detail.



Linnebo starts by referencing some of Frege’s ideas:

The Fregean argument is based on two premises, the first of which concerns the semantics of the language of mathematics:

Classical Semantics.
The singular terms of the language of mathematics purport to refer to mathematical objects, and its first-order quantifiers purport to range over such objects.

Most sentences accepted as mathematical theorems are true (regardless of their syntactic and semantic structure).

These premises are worded in complicated ways, but they boil down to simple logic:

1) Mathematical theorems are true.

2) Mathematical theorems refer to mathematical objects.

3) Therefore, mathematical objects exist.

The article goes into a little more detail (you can read more here), but this is the gist of it.

(Please note that the above logic is not intended to be sound. Instead, it is intended to facilitate the understanding of the idea of existence.)



Abstractness says that every mathematical object is abstract, where an object is said to be abstract just in case it is non-spatiotemporal and (therefore) causally inefficacious . . .

. . . For if these objects had spatiotemporal locations, then actual mathematical practice would be misguided and inadequate, since pure mathematicians ought then to take an interest in the locations of their objects, just as physicists take an interest in the locations of theirs.

The second of the three ideas, abstraction, is much less complicated than existence. If an object is abstract, it does not exist in space-time (also known as the material world). Other entities that are non-spatiotemporal may include ideas, thoughts, and concepts, depending on what philosophical outlook you have.

Abstract objects exist in an abstract world, sometimes thought of as a mirror to our own. This is one area of Platonism and mathematical Platonism differ. While Platonism states that the abstract world in the more fundamental/superior world to our own, mathematical Platonism does not assert this superiority.



Independence says that mathematical objects, if there are any, are independent of intelligent agents and their language, thought, and practices . . .

. . . had there not been any intelligent agents, or had their language, thought, or practices been different, there would still have been mathematical objects.

The last of the three ideas, independence, is perhaps the simplest idea of the three. Independence states that mathematical entities are more than a human construct, and that they exist independently of us. This means that they were discovered by humans instead of created by humans, which is an important distinction. What independence implies is that (if they exist), other conscious entities would also discover mathematics in a similar way to us, or at least the basic concepts would be the same.


Image taken from pixabay.com and used/modified under Creative Commons License.

To summarize, mathematical Platonism states that mathematical objects (such as 3 and π) exist, are non-spatiotemporal, and were discovered as opposed to created by humans. This theory for the explanation of the existence of mathematical objects makes the most sense to me – for as I discussed in my earlier blog post, numbers don’t really exist in the physical world (space-time). Five fingers are material objects and so are five sheep, but does five itself exist materially in the same manner? This theory offers what I see to be sound explanations for the properties of mathematical entities, helping to lay the foundations for the entire field of mathematics.