Hi oldman,
So do you really live something like 100 km NW of Durban? Must be interesting. I’ve had two cousins live in South Africa but never had a chance to visit myself.
You have a nice writing style too. I enjoy your symbolism.
I’d agree Penrose would opt for, or perhaps more appropriately would be adamant about, “discovered”. One can’t deny he’s one of the most brilliant mathematicians in the world so rather than throw my idiotic 2 cents in, I’ll look to see what Penrose has to say. Funny also that Mazur, although writing a paper that tries to portray the two sides without too much bias, also seems to be a Platonist. Or at least refuses to accept the anti-Platonist view.
The problem with the question however, is that it’s just too short. And the paper by Mazur, although spirited, doesn’t seem to really explain very well what is meant by “discover” and “invent”. Instead, his paper seems to assume you already know what the argument is all about. So I apologize for the length of this post, but I think we have to understand what is meant. For that I’ll digress momentarily and come around to try and explain my understanding of Penrose’s view, because I think it’s Penrose that really fleshes some of this out nicely.
Here, I’ll treat the word “physical” to mean
that which can be objectively measured and found to exist in 3 dimensions and that of time. In this sense, something which is physical is a subset of the natural world since there are other phenomena which exist that can’t be considered physical. <gasp! more in a moment..> So I’ll consider the word “natural” to mean everything which exists that is both objectively observed and subjectively observed.
- For the natural world, discovered means that which existed at all times.
- Invented means that which came into existence only because of happenstance.
This is a slightly different definition of the terms than might be used elsewhere so I’ll try and explain what is meant through definitions and examples. Hopefully, the reason for doing this will become clear momentarily. Note also, I think these definitions will better coincide with what Mazure, Penrose and others who’ve written on this topic want.
Different Discovered worlds:
1. Physical world: Physical, 4 dimensional world. Meets criteria for Discovered.
2. Mental world: (ex: redness of an apple, the tone of a musical note, the sweetness of sugar, the sensation of making a choice) Not objectively measurable, so it doesn’t fit into the physical world. Meets criteria for Discovered.
3. Platonic Mathematical world: Per Penrose, Mazure, others. But is it really discovered?
These are the ONLY “Discovered” worlds. We might discover some unknown species of microbe on Mars for example, but that isn’t what is meant by discovered by Mazur and Penrose. For example, Mazur states:
If we adopt the Platonic view that mathematics is discovered, we are suddenly in surprising territory, for this is a full-fledged theistic position. Not that it necessarily posits a god, but rather that its stance is such that the only way one can adequately express one’s faith in it, the only way one can hope to persuade others of its truth, is by abandoning the arsenal of rationality, and relying on the resources of the prophets.
…
For the Platonists. One crucial consequence of the Platonic position is that it views mathematics as a project akin to physics, Platonic mathematicians being – as physicists certainly are describers or possibly predictors (I THINK HE’S REFERRING TO THE “PROPHETS” HERE) – not, of course, of the physical world, but of some other more noetic entity. Mathematics – from the Platonic perspective- aims, among other things, to come up with the most faithful description of that entity.
I’ll quote Penrose momentarily, but it seems obvious from the context that both Penrose and Mazur have something other in mind than simply the ‘discovery’ of life on Mars.
I think that most would agree these are different ‘worlds’ but that isn’t to indicate that they can exist independent from each other. For example, we might assume the mental world and the mathematical world are supervenient on the physical world. That is, the mental world requires the physical world to exist. The mathematical world might also be seen to require the physical world to exist. One might also argue that the mathematical world however, can’t exist without the mental world, so perhaps the mathematical world requires a mental world, which requires a physical world. Penrose would seem to suggest however, that each of the three above “worlds” are interrelated, and although they may require each other to exist, Penrose suggests these are to be seen as ‘sets’ analogous to mathematical sets, which overlap but have parts which DON’T OVERLAP! How can that be and how does he argue this?
I think first, we need to examine some examples of ‘inventions’ to understand what exists and how they relate to the above 3 potential ‘worlds’. Examples of inventions:
1. Things made of matter or energy: Exist in physical world. Sailboats, cars, monkeys, mountains, planets and galaxies are all made from matter/energy and exist in time and space. Thus, they are all inventions of the physical world since any specific one of them came about only because of happenstance.
2. Stories: Although a story can be written in a book, and the book exists in the physical world, the story itself can only have meaning if a mind is contemplating it. The actual story is invented and exists in the mental world.
3. Music: Again, there can be sound pressure waves which are part of the physical world, but the music itself, just like any qualia, exists only in the mental world. Music meets criteria for “invented”.
4. Art: Same as musical, but physically may include other forms of interactions such as a clay sculpture or light (em waves). Art is generally made of something physical but the appreciation of it as “art” is mental. Art is an invention.
5. The academic pursuit of physics, engineering, biology, etc…: These are all ‘ideas’ or models about the physical world which require a mental world and a mathematical description. Physical laws and various physical interactions are all modeled by these various areas of science. These models should be considered interpretations of the physical world, so all of these are inventions of the mental world as a minimum. Our interpretations are inventions, despite the fact that what we are working with is real and exists in the physical world.
Penrose argues for a “Platonic world of absolute mathematical forms” possessed by the physical world.
The very question of the internal consistency of a scientific model, in particular, is one that requires that the model be precisely specified. The required precision demands that the model be a mathematical one, for otherwise one cannot be sure that these questions have well-defined answers.
If the model itself is to be assigned any kind of ‘existence’, then this existence is located within the Platonic world of mathematical forms. Of course, one might take a contrary viewpoint: namely that the model is itself to have existence only within our various minds, rather than to take Plato’s world to be in any sense absolute and ‘real’. Yet there is something important to be gained in regarding mathematical structures as having a reality of their own.
(pg 12, The Road to Reality)
Section 1.4 (pg 17) begins his discussion of “three worlds and three deep mysteries”. His Figure 1.3 can be found on the web here:
http://www.stefangeens.com/trinity.gif
In Figure 1.3, he shows what are sets. The Platonic mathematical world has some subset which contains or is projected upon the physical world. There is a subset of the physical world which is contains the mental world. And there is a subset of the mental world which contains the Platonic mathematical world. About this, he writes:
It may be noted, with regard to the first of these mysteries – relating the Platonic mathematical world to the physical world- that I am allowing that only a small part of the world of mathematics need have relevance to the workings of the physical world. It is certainly the case that the vast preponderance of the activities of pure mathematicians today has no obvious connection with physics, nor with any other science, although we may be frequently surprised by unexpected important applications. Likewise, in relation to the second mystery, whereby mentality comes about in association with certain physical structures (most specifically, healthy, wakeful human brains), I am not insisting that the majority of physical structures need induce mentality. While the brain of a cat may indeed evoke mental qualities, I am not requiring the same for a rock. Finally, for the third mystery, I regard it as self-evident that only a small fraction of our mental activity need be concerned with absolute mathematical truth! … These three facts are represented in the smallness of the base of the connection of each world with the next, the worlds being taken in a clockwise sense in the diagram.
Thus, according to Fig. 1.3, the entire physical world is depicted as being governed according to mathematical laws.
Penrose suggests that the mathematical world is discovered and is every bit as real as the mental world which is every bit as real as the physical world, albeit, real in a different sense of the term. He’s stating it is discovered because although nature obeys mathematical laws, there are ‘mathematical laws’ which have no application to the physical world, and these laws can only have a basis if there exists a mental world to contemplate them.
Anyway, that’s what Penrose seems to be saying. Here’s just one more from U of Oregon:
Thus, there came into existence two schools of thought. One that mathematical concepts are mere idealizations of our physical world. The world of absolutes, what is called the Platonic world, has existence only through the physical world. In this case, the mathematical world is the same as the Platonic world and would be thought of as emerging from the world of physical objects.
The other school is attributed to Plato, and finds that Nature is a structure that is precisely governed by timeless mathematical laws. According to Platonists we do not invent mathematical truths, we discover them. The Platonic world exists and physical world is a shadow of the truths in the Platonic world. This reasoning comes about when we realize (through thought and experimentation) how the behavior of Nature follows mathematics to an extremely high degree of accuracy. The deeper we probe the laws of Nature, the more the physical world disappears and becomes a world of pure math.
Mathematics transcends the physical reality that confronts our senses. The fact that mathematical theorems are discovered by several investigators indicates some objective element to mathematical systems. Since our brains have evolved to reflect the properties of the physical world, it is of no surprise that we discover mathematical relationships in Nature.
The laws of Nature are mathematical mostly because we define a relationship to be fundamental if it can be expressed mathematically.
Ref: http://abyss.uoregon.edu/~js/ast221/lectures/lec01.html
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