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Is mathematics real or not?

Four scientists weigh in, on behalf of the Japanese Kavli Foundation:

Max Tegmark (pro real): It’s actually an old idea. Even Galileo exclaimed that the universe is a grand book written in the language of mathematics because he was wowed by all the astronomic regularities discovered in his time, for instance the precise circular or elliptical orbits of planets. Long afterwards, a whole set of subatomic particles were predicted by mathematical principles and then discovered, the Higgs boson being the most recent example. Even the possibility of curved shape of outer space was predicted centuries earlier by non-Euclidian geometry. So nature is clearly giving us hints that the universe is mathematical. I’ve taken it to the extreme by proposing that our entire physical reality isn’t just described by math, but that it is a mathematical structure, having no properties besides mathematical properties.

Brian Butterworth (anti-real): We’ve pinpointed an area of the human brain where there’s a specialist neural network that responds to counting the number of objects in a set. This area of the brain can recognize numbers across modalities. In other words, it can recognize three cats, three tones or three wishes. A similar area in the brain of the monkey does the same job. We even discovered the guppy — a small fish with a tiny brain — has one system for detecting small sets of up to four objects and one for larger sets. My argument is that we evolved a brain-based system for detecting and comparing the number of items in groups. Humans have developed symbolism for these numbers and elaborated on them to create the kinds of mathematics that Max and Simeon need to describe the universe. Numbers are not necessarily a property of the universe, but rather a very powerful way of describing some aspects of the universe.

TKF: (moderator)So you disagree with Dr. Tegmark’s notion that electrons are merely numbers?

Brian Butterworth (anti-real):Yes, because in order to have a physical explanation for phenomena, you have to have a cause for it. But how can a number be a cause? It’s true that you can use numbers to describe electron properties, but that doesn’t mean those numbers are actually a property of that physical object. Twoness is a property of a set of objects, such as two cups, or two electrons. But it is independent of the kinds of objects that are in the set for which it is a property. A set of two cups is different from a set of two electrons so twoness can’t have the same causal property for cups and electrons.

Why not? Thoughts? More.

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18 Responses to Is mathematics real or not?

  1. 0 = 1 + e^ i*pi

    . . . Euler.

    KF

  2. 2
    Kantian Naturalist

    As I read Metaphysics, “numbers cannot be a cause” is actually central to Aristotle’s objection to Plato’s account of the Forms. (Better put: we cannot comprehend how numbers could be a cause.) These problems have been with us for a long, long time!

    I think the anti-realist argument as put here is pretty vulnerable to realist objections. The realist isn’t necessarily committed to saying “twoness is a cause”, but to saying that reality has the right sort of structure to be describable by mathematics.

    And she can say that just as we have perceptual capacities that reliably detect the presence of visible things, and touchable things, and so, we also have the conceptual capacities that reliably detect abstract structures and patterns.

    I think that the whole history of nominalism-cum-skepticism shows that it’s exceedingly difficult — the Platonic realist would say “impossible!” but who knows? — to develop a fully coherent, consistent view that says that our perceptual capacities reliably put us in cognitive contact with concrete objects, but that our conceptual capacities do not reliably puts us in cognitive contact with abstract objects.

    Any view that undermines the latter will, sooner or later, undermine the former (Hume); any view that affirms the latter will, sooner or later, affirm the latter (Plato, Aristotle).

    My main objection to Platonic realism is that it requires mind-body dualism, and that falls afoul of the interaction problem. But I’ll readily admit that a nominalistic treatment of mathematics, while nicely consistent with materialism, is unsatisfying as an account of mathematics.

  3. Kairos,

    I was going to comment on your Euler post but comments were turned off.

    In particular, I was going to address the idea of “freethinking.” The freethinkers are not atheist materialists, who are slavethinkers, still mired in decaying 19th-century pseudoscience. The real freethinkers are ID proponents, who are following the evidence where it leads.

    Here’s one of my UD posts on this topic from 2011, which inspired 686 comments:

    http://www.uncommondescent.com.....ethinking/

  4. Well, where to start?, Let’s see, leading quantum physicist Anton Zeilinger has followed in John Archibald Wheeler’s footsteps (1911-2008) by insisting reality, at its most foundational level, is ‘information’.

    “It from bit symbolizes the idea that every item of the physical world has at bottom – at a very deep bottom, in most instances – an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes-no questions and the registering of equipment-evoked responses; in short, that things physical are information-theoretic in origin.”
    John Archibald Wheeler

    Why the Quantum? It from Bit? A Participatory Universe?
    Excerpt: In conclusion, it may very well be said that information is the irreducible kernel from which everything else flows. Thence the question why nature appears quantized is simply a consequence of the fact that information itself is quantized by necessity. It might even be fair to observe that the concept that information is fundamental is very old knowledge of humanity, witness for example the beginning of gospel according to John: “In the beginning was the Word.”
    Anton Zeilinger – a leading expert in quantum teleportation:
    http://www.metanexus.net/archi.....linger.pdf

    Materialism had postulated for centuries that everything reduced to, or emerged from material atoms, yet the correct structure of reality is now found by science to be as follows:

    1. material particles (mass) normally reduces to energy (e=mc^2)
    2. energy and mass both reduce to information (quantum teleportation)
    3. information reduces to consciousness (geometric centrality of conscious observation in universe dictates that consciousness must precede quantum wave collapse to its single bit state)

    Here are my references for the claim that mass “normally” reduces to energy:

    The reduction of matter to energy is comparatively easy to accomplish as is demonstrated by nuclear/atomic bombs:

    Atomic Bomb Explosion – video
    http://www.youtube.com/watch?v=-22tna7KHzI

    *6.4 mg of mass converted to energy in Hiroshima A-bomb
    *4,400,000 Hiroshima A-bombs equivalent to one ounce of mass
    *1 drop of water equivalent to 10 Hiroshima A-bombs

    Whereas, on the other hand, to convert energy to matter is a far more difficult proposition:

    Why is it impossible, at this point in time, to convert energy into matter?
    Excerpt: “Particle accelerators convert energy into subatomic particles, for example by colliding electrons and positrons. Some of the kinetic energy in the collision goes into creating new particles.
    It’s not possible, however, to collect these newly created particles and assemble them into atoms, molecules and bigger (less microscopic) structures that we associate with ‘matter’ in our daily life. This is partly because in a technical sense, you cannot just create matter out of energy: there are various ‘conservation laws’ of electric charges, the number of leptons (electron-like particles) etc., which means that you can only create matter/anti-matter pairs out of energy. Anti-matter, however, has the unfortunate tendency to combine with matter and turn itself back into energy. Even though physicists have managed to safely trap a small amount of anti-matter using magnetic fields, this is not easy to do.
    Also, Einstein’s equation, Energy = Mass x the square of the velocity of light, tells you that it takes a huge amount of energy to create matter in this way. The big accelerator at Fermilab can be a significant drain on the electricity grid in and around the city of Chicago, and it has produced very little matter.
    http://imagine.gsfc.nasa.gov/d.....0724a.html

    Moreover, it is important to note that a simple atom is certainly not ‘simple’:

    Delayed time zero in photoemission: New record in time measurement accuracy – June 2010
    Excerpt: Although they could confirm the effect qualitatively using complicated computations, they came up with a time offset of only five attoseconds. The cause of this discrepancy may lie in the complexity of the neon atom, which consists, in addition to the nucleus, of ten electrons. “The computational effort required to model such a many-electron system exceeds the computational capacity of today’s supercomputers,” explains Yakovlev.
    http://www.physorg.com/news196606514.html

    Yet somehow, serendipitously, shortly after the big bang, and in the nucleosynthesis of stars, all the pieces of the puzzle ‘spontaneously’ fell together to get these complex atoms to form ‘spontaneously’ from energy (at least according to atheists it had to be ‘spontaneous’):

    Big Bang
    After its (The Big Bangs) initial expansion from a singularity, the Universe cooled sufficiently to allow energy to be converted into various subatomic particles, including protons, neutrons, and electrons.
    http://en.wikipedia.org/wiki/Big_Bang

    The Elements: Forged in Stars – video
    http://www.metacafe.com/watch/4003861

    “Dr. Michael Denton on Evidence of Fine-Tuning in the Universe” (Remarkable balance of various key elements for life) – podcast
    http://intelligentdesign.podom.....3_59-07_00

  5. Here are my references for the claim that “energy and mass both reduce to information”:

    Ions have been teleported successfully for the first time by two independent research groups
    Excerpt: In fact, copying isn’t quite the right word for it. In order to reproduce the quantum state of one atom in a second atom, the original has to be destroyed. This is unavoidable – it is enforced by the laws of quantum mechanics, which stipulate that you can’t ‘clone’ a quantum state. In principle, however, the ‘copy’ can be indistinguishable from the original (that was destroyed),,,
    http://www.rsc.org/chemistrywo.....ammeup.asp

    Atom takes a quantum leap – 2009
    Excerpt: Ytterbium ions have been ‘teleported’ over a distance of a metre.,,,
    “What you’re moving is information, not the actual atoms,” says Chris Monroe, from the Joint Quantum Institute at the University of Maryland in College Park and an author of the paper. But as two particles of the same type differ only in their quantum states, the transfer of quantum information is equivalent to moving the first particle to the location of the second.
    http://www.freerepublic.com/fo.....1769/posts

    Physicists set new record for quantum teleportation with matter qubits – Apr 16, 2013
    Excerpt: “The greatest significance of our work is the dramatic increase in efficiency compared to previous realizations of matter-matter teleportation,” Nölleke said. “Besides, it is the first demonstration of matter-matter teleportation between truly independent systems and constitutes the current record in distance of 21 m. The previous record was 1 m.”
    http://phys.org/news/2013-04-p.....ubits.html

    How Teleportation Will Work -
    Excerpt: In 1993, the idea of teleportation moved out of the realm of science fiction and into the world of theoretical possibility. It was then that physicist Charles Bennett and a team of researchers at IBM confirmed that quantum teleportation was possible, but only if the original object being teleported was destroyed. — As predicted, the original photon no longer existed once the replica was made.
    http://science.howstuffworks.c.....ation1.htm

    Quantum Teleportation – IBM Research Page
    Excerpt: “it would destroy the original (photon) in the process,,”
    http://researcher.ibm.com/view_project.php?id=2862

    Unconditional Quantum Teleportation – abstract
    Excerpt: This is the first realization of unconditional quantum teleportation where every state entering the device is actually teleported,,
    http://www.sciencemag.org/cont.....6.abstract

    It is also very interesting to note that the quantum state of a photon is actually defined as ‘infinite information’ in its uncollapsed quantum wave state:

    Quantum Computing – Stanford Encyclopedia
    Excerpt: Theoretically, a single qubit can store an infinite amount of information, yet when measured (and thus collapsing the Quantum Wave state) it yields only the classical result (0 or 1),,,
    http://plato.stanford.edu/entr.....tcomp/#2.1

    Explaining Information Transfer in Quantum Teleportation: Armond Duwell †‡ University of Pittsburgh
    Excerpt: In contrast to a classical bit, the description of a (photon) qubit requires an infinite amount of information. The amount of information is infinite because two real numbers are required in the expansion of the state vector of a two state quantum system (Jozsa 1997, 1) — Concept 2. is used by Bennett, et al. Recall that they infer that since an infinite amount of information is required to specify a (photon) qubit, an infinite amount of information must be transferred to teleport.
    http://www.cas.umt.edu/phil/fa.....lPSA2K.pdf

    Of related note:

    “It always bothers me that in spite of all this local business, what goes on in a tiny, no matter how tiny, region of space, and no matter how tiny a region of time, according to laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out. Now how can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one stinky tiny bit of space-time is going to do?
    - Richard Feynman – one of the founding fathers of QED (Quantum Electrodynamics)
    Quote taken from the 6:45 minute mark of the following video:
    https://www.youtube.com/watch?v=obCjODeoLVw

    I don’t know about Feynman, but as for myself, being a Christian Theist, I find it rather comforting to know that it takes an ‘infinite amount of logic to figure out what one stinky tiny bit of space-time is going to do’:

    John1:1
    In the beginning was the Word, and the Word was with God, and the Word was God.

    of note: ‘the Word’ in John1:1 is translated from ‘Logos’ in Greek. Logos is the root word from which we derive our modern word logic
    per: etymonline.com

    THE INFINITY PUZZLE: Quantum Field Theory and the Hunt for an Orderly Universe.
    Excerpt: In quantum electrodynamics, which applies quantum mechanics to the electromagnetic field and its interactions with matter, the equations led to infinite results for the self-energy or mass of the electron. After nearly two decades of effort, this problem was solved after World War II by a procedure called renormalization, in which the infinities are rolled up into the electron’s observed mass and charge, and are thereafter conveniently ignored. Richard Feynman, who shared the 1965 Nobel Prize with Julian Schwinger and Sin-Itiro Tomonaga for this breakthrough, referred to this sleight of hand as “brushing infinity under the rug.”
    per: American Scientist

  6. Here are my references for the claim that “information reduces to consciousness”:

    The argument for God from consciousness can now be framed like this:

    1. Consciousness either preceded all of material reality or is a ‘epi-phenomena’ of material reality.
    2. If consciousness is a ‘epi-phenomena’ of material reality then consciousness will be found to have no special position within material reality. Whereas conversely, if consciousness precedes material reality then consciousness will be found to have a special position within material reality.
    3. Consciousness is found to have a special, even central, position within material reality.
    4. Therefore, consciousness is found to precede material reality.

    Four intersecting lines of experimental evidence from quantum mechanics that shows that consciousness precedes material reality (Leggett’s Inequalities, Wheeler’s Delayed Choice; Wigner’s Quantum Symmetries; Quantum Zeno effect)
    https://docs.google.com/document/d/1G_Fi50ljF5w_XyJHfmSIZsOcPFhgoAZ3PRc_ktY8cFo/edit

    The Galileo Affair and Life/Consciousness as the true “Center of the Universe”
    https://docs.google.com/document/d/1BHAcvrc913SgnPcDohwkPnN4kMJ9EDX-JJSkjc4AXmA/edit

    The Center Of The Universe Is Life – General Relativity, Quantum Mechanics, Entropy and The Shroud Of Turin – video
    http://vimeo.com/34084462

    Of related note: The following site is of related interest to establishing the claim of “consciousness preceding ‘material’ reality” through geometric considerations:

    The Scale of The Universe – Part 2 – interactive graph (recently updated in 2012 with cool features)
    http://htwins.net/scale2/scale.....olor=white

    The preceding interactive graph points out that the smallest scale visible to the human eye (as well as a human egg) is at 10^-4 meters, which ‘just so happens’ to be directly in the exponential center of all possible sizes of our physical reality (not just ‘nearly’ in the exponential center!). i.e. 10^-4 is, exponentially, right in the middle of 10^-35 meters, which is the smallest possible unit of length, which is Planck length, and 10^27 meters, which is the largest possible unit of ‘observable’ length since space-time was created in the Big Bang, which is the diameter of the universe. This is very interesting for, as far as I can tell, the limits to human vision (as well as the size of the human egg) could have, theoretically, been at very different positions than directly in the exponential middle;

    Of course there is a lot more that has and can be said by people much more qualified than I to speak on the subject, but in my limited capacity I find the universe to be far more friendly to Theistic presupposition, particularly Christian Theistic presuppositions, than many physicists realize or are willing to admit.

    Music and Verse:

    Kari Jobe – Revelation Song – Passion 2013
    http://www.youtube.com/watch?v=3dZMBrGGmeE

    Hebrews 1:3
    The Son is the radiance of God’s glory and the exact representation of his being, sustaining all things by his powerful word.

  7. 1. I always have to laugh when someone says we have, “…pinpointed an area of the human brain…”. What they mean is that when someone actualizes data this area of the brain has more activity. They use this proxy of activity in a certain area as evidence against dualism. But they really have no idea what the nerves are doing, or how the meaning of a number is comprehended.

    2. A number can be a cause if I mentally assign it to be a cause.

    For example, when I play a game of monopoly whether I go forward 5 or 6 spaces may make the difference between paying $75 to the bank or $2000 to the owner of boardwalk.

    Or for another example, when someone asks me to to a task in 10 minutes, what is the cause of me eventually doing it in 10 minutes. It is the number 10. The reason I believe the number ( or maybe – my perception of the number ) to be the cause is that the number 10 can be communicated to me in many different ways — orally ( in many different languages ), by writing it down, by tapping that many times ( making me both count, and then respond ). It is the number 10, not the specific material of media through which the number is communicated, that causes the response.

    Materialists really have no answer for how human beings can perform delayed future actions after an arbitrary time. We know this happens, but they can not come up with a reasonable description of how it happens without introducing the concept of libertarian free will.

  8. It is hard to find a less important question than whether numbers (or other mathematical objects) are real.

    Personally, I’m a fictionalist. I’ve had people tell me that mathematics can’t work with fictionalism, or can’t be used in science. What nonsense. It works the same for me as for anybody else. The question of whether numbers are real does not arise.

  9. Neil Rickert, I have a cartoon for you:

    Calvin and Hobbes – cartoon – The Mathematical Atheist
    http://s3.hubimg.com/u/270622_f520.jpg

    as to your overall belief, all I can say is that One man’s trash is another man’s treasure:

    The Unreasonable Effectiveness of Mathematics in the Natural Sciences – Eugene Wigner – 1960
    Excerpt: ,,certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess.,,,
    It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them.,,,
    The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
    http://www.dartmouth.edu/~matc.....igner.html

    Mathematics and Physics – A Happy Coincidence? – William Lane Craig – video
    http://www.metacafe.com/w/9826382

    Kurt Gödel – Incompleteness Theorem – video
    http://www.metacafe.com/w/8462821

    BRUCE GORDON: Hawking’s irrational arguments – October 2010
    Excerpt: Rather, the transcendent reality on which our universe depends must be something that can exhibit agency – a mind that can choose among the infinite variety of mathematical descriptions and bring into existence a reality that corresponds to a consistent subset of them. This is what “breathes fire into the equations and makes a universe for them to describe.” Anything else invokes random miracles as an explanatory principle and spells the end of scientific rationality.,,,
    Universes do not “spontaneously create” on the basis of abstract mathematical descriptions, nor does the fantasy of a limitless multiverse trump the explanatory power of transcendent intelligent design. What Mr. Hawking’s contrary assertions show is that mathematical savants can sometimes be metaphysical simpletons. Caveat emptor.
    http://www.washingtontimes.com.....arguments/

    Nature by Numbers – The Fingerprint of God – video
    https://vimeo.com/9953368

    What pi sounds like when put to music – cool video
    http://www.youtube.com/watch?v=YOQb_mtkEEE

    “It appears that the Creator shares the mathematicians’ sense of beauty.”
    - Alexander Vilenkin (stated in reference to Euler’s Identity in particular)
    http://rfforum.websitetoolbox.com/post?id=3754268

  10. Personally, I’m a fictionalist. I’ve had people tell me that mathematics can’t work with fictionalism, or can’t be used in science.

    Think of a Number

  11. What nonsense, indeed!

    We perceive duality (and a lot of other concepts attributed to what we call mathematics) because our senses evolved or were created to perceive practical information for our survival as an integral part of the universe.

    “Oh,” some people might argue, “but you can see double when it’s not true, therefore our senses are *unreliable*, and everything could be an illusion. Thus, no one (especially me) can be held responsible for anything they do by anyone else (especially God).”

    “Oh,” God might observe, “you lived in all other ways as if you could and did trust your senses as reality, so I’ll simply judge you by your own standards.”

    Oops.

    However, I’m not so keen on the significance of the middle of things (including dynamic range). I think Sagan was extremely impressed that sun wasn’t the center of the universe (as well as our earth being a pale blue dot from some arbitrary vantage point in space, and with numbers with a lot of zeros in them). As everyone with a belly button knows, the middle of anything is of *prime* importance! Not.

    On the other hand, we do seem to be on a “privileged planet” with respect to observational astronomy–including our place in Time, considering the current apparent sizes of the sun and moon.

  12. Math is NOT real. The divisions we call math are just a language of the reality. our language is just sounds organized to expressed thoughts. Yet our thoughts are not sounds.
    Math has nothing to do with the universe. Its in fact just that the universe is so well ordered that divisions can produce a representation of this order and finely ordered indeed.
    Numbers only represent things. They are not the essence of things.
    It could only be a natural ordered universe would be measurable.
    Math is a measuring of order. It does not exist without the order.
    Math is not real. The order of the universe is real. Mans invention of math is a primitive language of this sell done order.
    Biology probably has a greater order then physics. however putting it into math is still too complicated.
    Wild ideas like evolution proving this.

  13. Groan. Is this the future of ID ? Discussing how many angels on the head of a pin ?

    At least its a break from the usual theology, I guess.

  14. Gil Dodgen;

    The real freethinkers are ID proponents, who are following the evidence where it leads.

    Really? What is this evidence? Is there now a working theory of “Intelligent Design” that makes some testable prediction?

    This would be rather more interesting than whether numbers are real or not.

    On topic:

    Numbers are complex and thus can be real, imaginary or both!

  15. Shoot, apparently some atheists are not even sure if they are real, much less if math is:

    Why Intellectuals Laugh at Atheists (Part 3) – video
    http://www.youtube.com/watch?v=eaBMtiXaoWI

    semi-related notes:

    Why No One (Can) Believe Atheism/Naturalism to be True – video
    Excerpt: “Since we are creatures of natural selection, we cannot totally trust our senses. Evolution only passes on traits that help a species survive, and not concerned with preserving traits that tell a species what is actually true about life.”
    Richard Dawkins – quoted from “The God Delusion”
    http://www.youtube.com/watch?v=N4QFsKevTXs

    Scientific Peer Review is in Trouble: From Medical Science to Darwinism – Mike Keas – October 10, 2012
    Excerpt: Survival is all that matters on evolutionary naturalism. Our evolving brains are more likely to give us useful fictions that promote survival rather than the truth about reality. Thus evolutionary naturalism undermines all rationality (including confidence in science itself). Renown philosopher Alvin Plantinga has argued against naturalism in this way (summary of that argument is linked on the site:).
    Or, if your short on time and patience to grasp Plantinga’s nuanced argument, see if you can digest this thought from evolutionary cognitive psychologist Steve Pinker, who baldly states:
    “Our brains are shaped for fitness, not for truth; sometimes the truth is adaptive, sometimes it is not.”
    Steven Pinker, evolutionary cognitive psychologist, How the Mind Works (W.W. Norton, 1997), p. 305.
    http://blogs.christianpost.com.....ism-12421/

    Around the 12:00 minute mark of the following video Pastor Joe Boot reflects on the self-defeating nature of the atheistic worldview:

    Defending the Christian Faith – Pastor Joe Boot – video
    http://www.youtube.com/watch?v=wqE5_ZOAnKo

  16. Alan Fox:

    What is this evidence?

    It has been presented and you choked on it, as usual.

    Is there now a working theory of “Intelligent Design” that makes some testable prediction?

    Yes, however it is obvious that your position doesn’t have anything like that.

  17. Really? What is this evidence?

    Computer programs are designed. Living systems include computer programs, far more sophisticated than anything designed by the most intelligent humans.

    This is why I no longer waste my time arguing with flat-earthers and other science-deniers who ask for evidence of design in living systems.

  18. A more interesting question is whether mathematicians are real or not.

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