James Robert Brown: The Continuum Hypothesis

James Robert Brown FRSC is a Canadian philosopher of science. He is an emeritus Professor of philosophy at the University of Toronto. In the philosophy of mathematics, he has advocated mathematical Platonism, visual reasoning, and in the philosophy of science he has defended scientific realism mostly against anti-realist views associated with social constructivism.

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00:00:00 Introduction
00:01:17 The connection between mathematics and ethics
00:03:05 Mathematical concepts that are interpreted solely physically (thick and thin concepts)
00:16:33 The continuum hypothesis
00:19:23 A counter proof to the continuum hypothesis
00:24:45 What's possible can have probability of zero (measure zero, technically)
00:33:50 Platonism and "thick" mathematical concepts
00:34:39 Moral realism / objectivity, without God
00:36:59 Does 2+2=5?
00:40:43 Moral intuitions as serving a marker for what's "correct"?
00:46:42 What's makes some theory correct beside its ability to predict / explain?
00:47:54 Does libertarian Free Will exist, and how is it coexistent with physics?
00:53:13 VIEWER TEST: Platonism test for the audience
00:54:53 Did Shakespeare invent Hamlet, or discover it?
00:56:09 What created the Platonic world?
00:57:42 Eternal vs Sempiternal
00:59:12 Thought experiments as a tool of probing physics, without experiment
01:00:39 Galileo's thought experiment demonstrating objects fall at the same rate despite different masses / heaviness
01:03:39 Thought experiment demonstrating relative motion (invariance of physical laws given uniform motion)
01:05:01 The Tower thought experiment demonstrating the opposite of the above
01:09:43 VIEWER TEST: Are you a Newtonian absolutist, or a Leibnizian relationalist?
01:12:45 Why did Prof Brown go into the philosophy of math, instead of directly into mathematics (or physics)?
01:13:20 Have any philosophical problems ever been solved?
01:14:47 Is God good? Or is goodness independent of God?
01:15:41 Descartes vs Leibniz on God's power (God can do anything -- except what's logically impossible)
01:18:43 The parochial view of physicists / mathematicians to dismiss what they can't define
01:22:31 Thought experiment from Newton regarding the necessity of space
01:24:45 Einstein's variation on the above thought experiment
01:28:17 How to classify thought experiments
01:31:35 Prof. Brown's thoughts on Wolfram's TOE and Eric Weinstein's TOE
01:32:40 Nicholas Gisin's thoughts on real numbers, and free will
01:33:02 What other foundations are there to physics, other than classical logic?
01:39:05 What does Mathematical Realism look like when it's NOT Platonic?
01:45:31 The physical laws themselves as abstract entities, which have causal power
01:55:55 Lee Smolin's Principle of Precedence as a bridge between Platonism and non-Platonism
01:59:01 Deriving an "ought" from an "is"
02:03:29 What does James not like about Sam Harris?
02:06:45 Is Math discovered or invented?
02:10:18 Does Platonism entail some idea of a God?
02:14:49 Theism vs Deism
02:16:34 On the Sokal Affair and the trouble with Postmodernism
02:22:01 Limits of free speech?
02:27:41 The problem of commercializing research
02:33:09 On Jordan Peterson's "Darwinian" definition of truth

Subscribe if you want more conversations on Theories of Everything, Consciousness, Free Will, God, and the mathematics / physics of each.

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[andremiguel]

Curt, this is the first podcast I watch from you and I think you are doing a great job at interviewing these brilliant people. Better than Lex Fridman for example. You are really trying to push and test the guest assumptions and really trying to understand, not just wanting to sound smart by hearing your own voice going on monologues.
I am also a Mathematics/Philosophy guy so I find your channel incredible for my interests.
Keep the good work.

[TheoriesofEverything]

If you're interested in finding out what type of world view you have, then skip to:
00:53:13 VIEWER TEST: Platonism test for the audience
01:09:43 VIEWER TEST: Are you a Newtonian absolutist, or a Leibnizian relationalist?

[Achrononmaster]

@55:00 the point is the number for the play _Hamlet_ is an encoding, it does not generate the play unless you decode. So discovering the number is not the same as creating the play. It's a hardware + software problem. Not just a software problem. But it is more. The play _Hamlet_ has zero meaning without minds that comprehend the abstractions, and the integer encoding does not give you that, it does not give you "What it is like to have written, or viewed, the play." Never confuse information for knowledge, they are fundamentally different categories.
By throwing out some redundant bits it is possible to take the integer that encodes _Hamlet_ and simultaneously encode _Dr Seuss's ABC_ since the latter is a lot shorter. In other words, the integer is not the play.

[stephenbrown-okruhlik]

I am very grateful for all the comments, both during the initial presentation and after. I cannot reply to all, but I will try a couple, which might be helpful.

1. How could a zero probability event still be possible? You can get the idea by thinking of the probability of an even as 1/n, when there are n possible outcomes. If a raffle has 1000 tickets, then the prob of any one of them is 1/1000. If the raffle increases in size to infinitely many tickets, then the prob of any ticket goes to 0. However, some ticket will still be picked. Hence, prob 0 events are still possible.

2. How could a dart point pick out a single real number? A physical dart could not, since it would be at least one atom thick at the point. The dart example is a thought experiment. It uses physical concepts but often in an idealized way. Some thought experiments can be duplicated as actual experiments, but not all. Here I will shamelessly plug one of my books: The Laboratory of the Mind.

3. Is there free will and doesn't quantum mechanics (QM) say something about it? I act as if I and everyone else has free will and hold myself and others responsible for our actions. I don't know if I am right or wrong, but I can't really operate well in the world if I try to think otherwise. Doesn't QM imply free will? No. The standard view of QM --- which I accept --- is that the world is fundamentally nondeterministic. That shows that determinism is false. But it not the same as free will. Imagine you have some sort of randomizer in your brain (say, you move left if a particle decays at time t and move right if it does not decay). Then your actions would not be deterministic, but you do not have free will since you are not in control in any meaningful way. Free will is not the same as nondeterminism.

4. I'm glad people find interesting the little test for absolutism vs relationalism in space and time. I find that about half the people I ask (philosophers, physicists, mathematicians, in fact everyone), about half side with Newton and half side with Leibniz. Almost everyone is a Newtonian or a Leibnizian on both questions (as were Newton and Leibniz themselves). But it is logically possible to differentiate. Lee Smolin, for instance, in his book Time Reborn, is a relationalist about space but a substantivalist (of a sort) about time. (By the way, substantivaism is a better term than absolutism when debating this issue. It means space or time are things in their own right. Absolutism covers a lot of different things.)

Finally, thanks to Curt for all the work he did. I'm very impressed and very grateful.

[scottdow]

I'm so happy to have found your videos - very much a fan. Great interview!

[giovannironchi]

For a proof of the consistency of the negation of the continuum hypothesis and ZFC (assuming that ZFC is consistent) you can look up "Sheaves in Geometry and logic" by Saunders MacLane and Ieke Moerdijk. In chapter 6 they contruct a topos in which you can interpret set theory and have an "intermediate object" between the "naturals" and the "reals".

[DarthVagen]

I wish all entertainment was as interesting as this.

[StephenPaulKing]

An excellent episode of an Outstanding podcast. I am skeptical of the dart proof because it assumes an Aleph_1 sharp point on the darts (in order to be able to hit one point on a line). I am also squeezy about the simple equivalence assumed between 0 and 1/infinity without specifying the cardinality of the infinity or the measure (ignoring constructability).
Additionally, I find the very idea of using a natural language as a ground for metaphysical exploration since the set of our experiences can be shown to be both NWF and only sparsely connected to sensorimotor grounding, and our reasoning is not only covered by Natural Language construction. I think that NLs are finite since the differences within any given NL, the 'accents' are collapsible to equivalence by finite processes. Unless we are going to appeal to the "finite cover" idea (of Compactness) as being omnipotent, we can not hope to cover our intuitions (a second order or "thick" concept!) using any NL.
Proof?
Our logical facts are theory dependent and there are uncountable (became an axiom can differ from another by a minimal member of at set with aleph_1 cardinality) many axioms from which self-consistent theories can be constructed (ignoring resource cost). Construction is the key and given the sparsity of constructible subsets of NWF sets we are required to define a measure on the subset that we are considering...

[Achrononmaster]

@47:00 c'mon guys. I don't think Spinozarian pantheism is coherent, but as a model it's not bad for starters. The parts cannot have properties absent from the whole. So just very generically All of Reality (physical universe plus whatever else, platonic and all) has to have mental properties and creative ability (agency and will). Just as basic self-evident postulates. Then what almost all classical religions teach is basically correct, with Principle of Sufficient Reason (outside of time now, there's no need for strict temporal causal ordering). Then you have, fully consistent with everything we know or infer, the idea of a universal cause of all things, and this is where our intuitions derive from. Sure, ethical and moral advancement _evolve in physical time_ in some sentient species due to natural evolution, but the ethical and moral principles were always there platonically in potential, just like innate mathematical comprehension, but these are different types. (I think a lot of people, including decent physicists, still do not really think about what "potential" means --- the ontological term, not the electromagnetic/field term).
Thing is, people hate religion not because it is religion, but because the institutions and people proclaiming radical beliefs are anti-religious. The proper ethical human innate intuition is that being kind and generous and seeking truth are "good". What religious institutions, the church, the clergy, the synagogue, the temples, preach is anti all the goodness (for a large part). So a lot of the organized religion is objectively anti-religion, that's why people hate it and scientists and physicalists get driven into the "safer" insanity of materialism --- because they're too stupid or indoctrinated with "scientism" to think of conceptually separating religion from anti-religion.
Ironically, "scientism" is like this too, it is anti-scientific. A lot of professional scientists go off spouting nonsense and pseudo-science, claiming to be in that capacity still speaking as scientists. The most generous way to characterise such people is that they might be innocent of their ignorance or frauds, one or the other, so I prefer to say they're just innocent.

[bruceylwang]

As I say, the Key word is “structure” (property).
The realistic fact is: physical and non-physical are structures so that they can exchange energy.
So, both cosmos and consciousness are structures, they are connected by structures and exchange energy all the time. Such as, Mind is a structure. Mind and body are connected and exchange energy all the time.
… E=mc2+…+… where mc2 is one term of Energy.
… Value and fact are structures, they are connected with Energy. And, this the Truth.
Unfortunately, up to now, not many scientists are looking into “Structural Model” direction. However, I believe they will in the future.

[Achrononmaster]

@52:00 it is plain false that there is an unbroken reductionist chain down to microphysics. Where have you ever seen that spelt out? It does not exist. You can metaphysically _believe_ that past states determine the present, but you can never prove that information from the future Cauchy hypersurfaces is not influencing the present. Causation only appears uni-directional in time because of our memory capacity being strongly coupled to past low entropy states, but that is not evidence in the slightest that top-down causation (and by implication retrocausality) is not acting. A single closed timelike curve (wormhole topology) would ruin all reductionist accounts.

[carlwilson]

The "objects" of mathematics are discovered, the language with which we point at them are invented. The objects are structures. The rules by which we manipulate the language reflect the structure.

[Achrononmaster]

@1:26:00 the Newton/Mach bucket is really about the nature of the vacuum, not "space" _per se._ Mach/Leibniz are just saying the vacuum is empty nothingness. Newton is saying it is "stuff". Newton did not have quantum mechanics so could not know about virtual particles, had he known he'd have triumphed earlier before Leibniz died, and Mach and Einstein would never have had a squabble. But the fascinating thing here is that of course we now still only understand inertia partially, we've got vacuum pair production, Casimir invariants, entanglement structure (non-trivial topology), and the Higgs, but no one can run exact computations of the effects of inertia, hence the mass gap Millennium Prize problem, and further unsolved issues.

[Achrononmaster]

@40:00 children intuit that a pile of 3 blocks left of a pile of 2 is different to the pile of 3 being on the right. But they know the whole collections are the same, so they have category theoretic intuitions above and beyond Peano Arithmetic intuitions, "equals" is not the same as "equivalent" or "isomorphic."

[Nameeejz]

Very interesting conversation!!
Its surprising how my instinct for space was Newtonian but time was Liebnizian.
Space holding nothing makes more sense than time being perceivable without change. Is there any reason that time would also have to be an absolute thing in itself? I can imagine that as the universe began to exist there was no change and that is why exactly at its' moment of starting it would be infinite in duration. After it has started, time starts to speed up because the relations between things would compound

[firstnamesurname]

Just for James and Curt,
I will use Sanskrit words and not respond to questions about it ...

Free Will.

In order to The Chit apprehend Entropy as transitions from Parabrahman to Maya ... we need Absolute Chaos or Chaotic Inertia for Splitting Parabrahman in infinite paths ....

Brahman is a phase transition between Parabrahman and Maya ...

Maya is a phase transition between Brahman and Absolute Noise ...

That means that - in Parabrahman, Free Will doesn't Exist at All but Exists as a Subjective Illusion ... Not Freedom at All.

In Brahman, There are Negentropy balances and subtle oscillations between absolute order and Absolute Chaos ... Therefore, some degrees of freedom compared to The previous state.

In Maya, We got inertial flows random oscillations of pseudo chaotic order towards absolute chaos.

Then, More degrees of freedom compared to the previous state.

And in Absolute Chaos, we get absolute noise/disorder and Absolute degrees of freedom

Transitions of phase between Maya and Absolute Chaos is what the Apes identify as Natural Inertial Physical Laws ...

In this Multiverse Brane, Transitions from Maya to Brahman are not common in humans specimens ... but The Artifacts of The function of phase is what makes the Apes experience an integrated Chit ...

Therefore, James and Curt, If You want to observe Parabrahman, You MUST to Absolute Renounce to any sense of Free Will ... but You must catch an Inertial flow towards Absolute Unification ... because, in Maya, to renounce to The degrees of freedom in The Chit don't imply that the Chit is going to transition towards Brahman because Maya on itself is layered over a quasi-stable chaotic background ...therefore, without an attractor towards Brahman, the surrendering outcomes are random and very likely noneffective ...

.... and That's the Reason behind Krishna( Vishnu) said¨: '...let everything aside and just absolutely surrender to Him ...'

... and Jesus said: '...Let everything aside and absolutely surrender to My Father ...'

While the Vedic structure aspires to a direct transition from The Chit into Parabrahman ...

The Christian view uses Jesus as an attractor to Brahman for securing the ascension into Parabrahman ... but they fear and reject Parabrahman ... getting them stuck in Maya ...

That's all that Thag want to say





देवद्विषां निगमवर्त्मनि निष्ठितानां
पूर्भिर्मयेन विहिताभिरद‍ृश्यतूर्भि: ।
लोकान् घ्नतां मतिविमोहमतिप्रलोभं
वेषं विधाय बहु भाष्यत औपधर्म्यम् ॥ ३७ ॥

[Overthinktank]

Yeah this is the best podcast, subject of a podcast and podcast host for me just because it is about theory of everything and most likely attract polymaths. We need more polymaths in fact we should all be polymath in the future and a.i. and robots as a slave because capitalism requires slavery.

[onseayu]

well....i wouldn't be worried about PC if i were in math. i might care a little more if my kid wanted to major in gender studies for tens of thousands of dollars a year though

[Achrononmaster]

@1:16:00 Prof Brown has that one wrong. Moral good is not independent from God. The argument it is, boils down to speciousness (looks correct but is false). So is the omnipotence argument. They confuse the import of counterfactuals. "If" God told you "murder of this innocent child is "good", " then it'd be stupid, true. But a valid conclusion is 'the thing that told you this is not God.' It is a violation of a type of copernican principle, or an egregious anthropomorphism, to suppose our innate concepts of "the good" are our own. No. Nature endows us with these innate concepts (see Chomsky) and even if you are not a deist or theist like Descartes or Leibniz, you could go for Spinozarean pantheism. In all cases, it is consistent that our moral sensibilities derive from God. One can reach Prof Brown's conclusion that morals are independent from God only by a circular argument, by assuming our innate moral sensibilities are our own (i.e., are independent from God.) So it's a specious argument that Plato gave, as I said, albeit a clever argument (but sciolistic).
It is a common cognitive bias to think innate capacities are somehow "natural", biological, and so are independent of God. At the very minimal level of sophistication Spinoza would say, "Well, Nature _is_ God." I'm not a Spinozarean, but it's a valid refutation of Plato. The theist has the refutation, "Well all Nature proceeds from God, the prime Cause."
On Omnipotence: Omnipotence is not definable, but any finite definition of partial omnipotence is obviously never "power to do what is logically impossible." Also, the "raining and not raining" at the same time thought experiment is just silly, that happens for real, unless you define "raining" as every single spot on the ground gets a drop of rain, or something like that. The better example is, "a rock so heavy even God cannot lift it." It is again a specious argument, because all it shows is that either God is an inconsistent notion or such a rock is inconsistent. You can take either horn, and obviously a sensible theist can take the superior. So there is no paradox.
All similar atheist arguments directly against "existence of God" fail on similar sorts of definitional problems and specious arguments, such as assuming some notion of "God" that no one else (of sane mind) ever really assumes is coherent. It's always easy to refute the notion that an incoherent concept "does not exist." They are useful arguments because they tell you what concepts of "God" etc are incoherent. But that's all they do.

[hamzariazuddin]

I love these podcasts soooo interesting. What books are best to go through for all these thought experiments Mr Brown uses to see where your intuitive mind leads you on Philosophical schools...its a very interesting way to unravel where your intuition leads you...