An abstract game of chess is independent of the colors and shapes of the pieces, and of whether its moves are described on a physically existing board, by stylized computer-rendered images or by so-called algebraic chess notation — it's still the same chess game. Analogously, a mathematical structure is independent of the symbols used to describe it.
Image: Courtesy of Max Tegmark. Life without baggage Above we described how we humans add baggage to our descriptions. Now let's look at the opposite: how mathematical abstraction can remove baggage and strip things down to their bare essence. When chess aficionados call the Immortal Game beautiful, they're not referring to the attractiveness of the players, the board or the pieces, but to a more abstract entity, which we might call the abstract game, or the sequence of moves.
Chess involves abstract entities different chess pieces, different squares on the board, etc. For example, one relation that a piece may have to a square is that the former is standing on the latter. Another relation that a piece may have to a square is that it's allowed to move there. There are many equivalent ways of describing these entities and relations, for example with a physical board, via verbal descriptions in English or Spanish, or using so-called algebraic chess notation.
So what is it that's left when you strip away all this baggage? What is it that's described by all these equivalent descriptions? The Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not from properties of its ultimate building blocks, but from the relations between these building blocks.
The external physical reality is therefore more than the sum of its parts, in the sense that it can have many interesting properties while its parts have no intrinsic properties at all. This crazy-sounding belief of mine that our physical world not only is described by mathematics, but that it is mathematics, makes us self-aware parts of a giant mathematical object.
As I describe in the book, this ultimately demotes familiar notions such as randomness, complexity and even change to the status of illusions; it also implies a new and ultimate collection of parallel universes so vast and exotic that all the above-mentioned bizarreness pales in comparison, forcing us to relinquish many of our most deeply ingrained notions of reality. Indeed, we humans have had this experience before, over and over again discovering that what we thought was everything was merely a small part of a larger structure: our planet, our solar system, our Galaxy, our universe and perhaps a hierarchy of parallel universes, nested like Russian dolls.
However, I find this empowering as well, because we've repeatedly underestimated not only the size of our cosmos, but also the power of our human mind to understand it. They'd been told beautiful myths and stories, but little did they realize that they had it in them to actually figure out the answers to these questions for themselves.
And that the secret lay not in learning to fly into space to examine the celestial objects, but in letting their human minds fly. When our human imagination first got off the ground and started deciphering the mysteries of space, it was done with mental power rather than rocket power. If you decide to read it, then it will be not only the quest of me and my fellow physicists, but our quest.
Known as "Mad Max" for his unorthodox ideas and passion for adventure, Max Tegmark's scientific interests range from precision cosmology to the ultimate nature of reality, all explored in his new popular book, "Our Mathematical Universe.
His work with the SDSS collaboration on galaxy clustering shared the first prize in Science magazine's "Breakthrough of the Year: Already a subscriber? This argument is not new. Physicists tend to be "closeted non-Platonists," he says, meaning they often appear Platonist in public. But when pressed in private, he says he can "often extract a non-Platonist confession.
So if mathematicians, engineers, and physicists can all manage to perform their work despite differences in opinion on this philosophical subject, why does the true nature of mathematics in its relation to the physical world really matter? The reason, Abbott says, is that because when you recognize that math is just a mental construct—just an approximation of reality that has its frailties and limitations and that will break down at some point because perfect mathematical forms do not exist in the physical universe—then you can see how ineffective math is.
And that is Abbott's main point and most controversial one : that mathematics is not exceptionally good at describing reality, and definitely not the "miracle" that some scientists have marveled at. Einstein, a mathematical non-Platonist, was one scientist who marveled at the power of mathematics. He asked, "How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?
In , the physicist and mathematician Eugene Wigner described this problem as "the unreasonable effectiveness of mathematics. But whereas Wigner and Einstein might be considered mathematical optimists who noticed all the ways that mathematics closely describes reality, Abbott pessimistically points out that these mathematical models almost always fall short.
What exactly does "effective mathematics" look like? Abbott explains that effective mathematics provides compact, idealized representations of the inherently noisy physical world.
Maths is effective when it delivers simple, compact expressions that we can apply with regularity to many situations. It is ineffective when it fails to deliver that elegant compactness. Math only has the illusion of being effective when we focus on the successful examples. But our successful examples perhaps only apply to a tiny portion of all the possible questions we could ask about the universe. Some of the arguments in Abbott's paper are based on the ideas of the mathematician Richard W.
Hamming, who in identified four reasons why mathematics should not be as effective as it seems. Although Hamming resigned himself to the idea that mathematics is unreasonably effective, Abbott shows that Hamming's reasons actually support non-Platonism given a reduced level of mathematical effectiveness. Here are a few of Abbott's reasons for why mathematics is reasonably ineffective, which are largely based on the non-Platonist viewpoint that math is a human invention:.
There have likely been millions of failed mathematical models, but nobody pays attention to them. For example, in the s when transistor lengths were on the order of micrometers, engineers could describe transistor behavior using elegant equations. Today's submicrometer transistors involve complicated effects that the earlier models neglected, so engineers have turned to computer simulation software to model smaller transistors.
A more effective formula would describe transistors at all scales, but such a compact formula does not exist. For example, we see the Sun as an energy source for our planet, but if the human lifespan were as long as the universe, perhaps the Sun would appear to be a short-lived fluctuation that rapidly brings our planet into thermal equilibrium with itself as it "blasts" into a red giant. From this perspective, the Earth is not extracting useful net energy from the Sun. When counting bananas, for example, at some point the number of bananas will be so large that the gravitational pull of all the bananas draws them into a black hole.
At some point, we can no longer rely on numbers to count. One alleged benefit of this plenitudinous view is in the epistemology of mathematics. If every consistent mathematical theory is true of some universe of mathematical objects, then mathematical knowledge will, in some sense, be easy to obtain: provided that our mathematical theories are consistent, they are guaranteed to be true of some universe of mathematical objects.
Colyvan and Zalta criticize it for undermining the possibility of reference to mathematical objects, and Restall , for lacking a precise and coherent formulation of the plenitude principle on which the view is based. Martin proposes that different universes of sets be amalgamated to yield a single maximal universe, which will be privileged by fitting our conception of set better than any other universe of sets.
In object theory, moreover, two abstract objects are identical just in case they encode precisely the same properties. Assume that object realism is true. For convenience, assume also Classical Semantics. These assumptions ensure that the singular terms and quantifiers of mathematical language refer to and range over abstract objects. Given these assumptions, should one also be a mathematical platonist? In other words, do the objects that mathematical sentences refer to and quantify over satisfy Independence or some similar condition?
It will be useful to restate our assumptions in more neutral terms. We can do this by invoking the notion of a semantic value , which plays an important role in semantics and the philosophy of language.
In these fields it is widely assumed that each expression makes some definite contribution to the truth-value of sentences in which the expression occurs. This contribution is known as the semantic value of the expression. It is widely assumed that at least in extensional contexts the semantic value of a singular term is just its referent. Our assumptions can now be stated neutrally as the claim that mathematical singular terms have abstract semantic values and that its quantifiers range over the kinds of item that serve as semantic values.
What is the philosophical significance of this claim? In particular, does it support some version of Independence? The answer will depend on what is required for a mathematical singular term to have a semantic value. It suffices for the term t to make some definite contribution to the truth-values of sentences in which it occurs. The whole purpose of the notion of a semantic value was to represent such contributions. It therefore suffices for a singular term to possess a semantic value that it makes some such suitable contribution.
This may even open the way for a form of non-eliminative reductionism about mathematical objects Dummett a, Linnebo Although it is perfectly true that the mathematical singular term t has an abstract object as its semantic value, this truth may obtain in virtue of more basic facts which do not mention or involve the relevant abstract object.
Compare for instance the relation of ownership that obtains between a person and her bank account. Although it is perfectly true that the person owns the bank account, this truth may obtain in virtue of more basic sociological or psychological facts which do not mention or involve the bank account.
If some lightweight account of semantic values is defensible, we can accept the assumptions of object realism and Classical Semantics without committing ourselves to any traditional or robust form of platonism.
We conclude by describing two further examples of lightweight forms of object realism that reject the platonistic analogy between mathematical objects and ordinary physical objects. First, perhaps mathematical objects exist only in a potential manner, which contrasts with the actual mode of existence of ordinary physical objects. According to Aristotle, the natural numbers are potentially infinite in the sense that, however large a number we have produced by instantiating it in the physical world , it is possible to produce an even larger number.
But Aristotle denies that the natural numbers are actually infinite: this would require the physical world to be infinite, which he argues is impossible. Following Cantor, most mathematicians and philosophers now defend the actual infinity of the natural numbers.
This is made possible in part by denying the Aristotelian requirement that every number needs to be instantiated in the physical world. When this is denied, the actual infinity of the natural numbers no longer entails the actual infinity of the physical world. No matter how many sets have been formed, it is possible to form even more.
If true, this would mean that sets have a potential form of existence which distinguishes them sharply from ordinary physical objects. Second, perhaps mathematical objects are ontologically dependent or derivative in a way that distinguishes them from independently existing physical objects Rosen , Donaldson For example, on the Aristotelian view just mentioned, a natural number depends for its existence on some instantiation or other in the physical world.
There are other versions of the view as well. For example, Kit Fine and others argue that a set is ontologically dependent on its elements. This view is also closely related to the set-theoretic potentialism mentioned above. I gratefully acknowledge their support. What is Mathematical Platonism? The Fregean Argument for Existence 2. Objections to Mathematical Platonism 3.
Between object realism and mathematical platonism 4. Mathematical platonism can be defined as the conjunction of the following three theses: Existence. There are mathematical objects. Mathematical objects are abstract. Morton and S. Stich, eds. Benacerraf, Paul and Putnam, Hilary eds. Second edition. Burgess, John P. Cole, Julian C.
Pataut ed. Colyvan, Mark and Zalta, Edward N. Feferman et al, ed. III, — Benacerraf and H. Putnam, eds. Reprinted in Hale and Wright Hersh, Reuben, , What is Mathematics, Really?
Linsky, Bernard and Zalta, Edward N. Martin, Donald A. Butts and J. Hintikka eds. Quine, W. Rees, D. Polkinghorne ed. Academic Tools How to cite this entry. Enhanced bibliography for this entry at PhilPapers , with links to its database.
Other Internet Resources [Please contact the author with suggestions. Related Entries abstract objects mathematics, philosophy of: indispensability arguments in the mathematics, philosophy of: naturalism physicalism Plato: middle period metaphysics and epistemology. The science that draws necessary conclusions. Symbolic logic. The study of structures. The account we give of the timeless architecture of the cosmos. The poetry of logical ideas. Statements related by very strict rules of deduction.
A means of seeking a deductive pathway from a set of axioms to a set of propositions or their denials. A proto-text whose existence is only postulated.
A precise conceptual apparatus. The study of ideas that can be handled as if they were real things. The manipulation of the meaningless symbols of a first-order language according to explicit, syntactical rules. A field in which the properties and interactions of idealized objects are examined.
The science of skillful operations with concepts and rules invented for the purpose. Conjectures, questions, intelligent guesses, and heuristic arguments about what is probably true. The longest continuous human thought. Laboriously constructed intuition.
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