It would be contradictory if geometry was different for each different person, since then we would be unable to share a world and communicate at all. — Agustino

That doesn't follow. The difference between a perfectly flat space and one that is curved very, very slightly would make no difference at all to the ability to communicate.

Oh, do you mean the OP is just your opinion? In that case there's nothing to argue about. It would be very presumptuous of anybody to tell you that what you think is your opinion is not really your opinion.

I like the Guardian but I wouldn't use it as a source in a debate because, although it has enormously greater integrity than the Mail, gun enthusiasts can too easily accuse it of left bias. Whether just or not, such accusations do derail discussions. Best to get something from an academic study or a government statistics agency.

I wasn't sure what to make of the Guardian article either. Most of it was just about 'police recorded crimes' of all types, not specific to violent crime or manufactured weapons.

The second para refers to increase in gun crime of 20% over the preceding year but, for all we know, that's just part of the usual year to year variation, rather than evidence of a sustained trend. In any case, the increase would need to be in the order of thousands of per cent for Britain's per capita gun crime casualty rate to near that of the US.

I see you've just added a link to a government statistics report. Good idea!
The numbers match the Guardian report.

I have seen no convincing evidence that judaism does not believe in some sort of afterlife existence, therefore, I am skeptical about that claim. — CuddlyHedgehog

That relates to a claim about one religion that was made part way down the thread. But you still have your OP sitting there consisting of two unsupported claims, covering all religions, not just one. A reader of this thread is entitled to be skeptical of your claims, and you have done nothing to dispel that skepticism.

Now, if I may, I shall remain skeptical of your argument. — CuddlyHedgehog

Skepticism is inimical to your position in this situation. You have made two claims in the OP. The skeptical position is to doubt any claim. Your claims remain under a cloud of skeptical doubt unless you can find some convincing evidence to dispel that cloud.

The promise of life after death is religion's lure. — CuddlyHedgehog

For some adherents of some religions, sure. But for all adherents of all religions? I doubt it.

If we had the consciousness of its necessity, then we couldn't be wrong, could we? — Agustino

This sounds an interesting avenue to explore. As I recall our discussion of a few days ago, you are conscious of its necessity, and I am conscious of its non-necessity (importantly, that is not the same thing as being conscious of the necessity of its negation!).

The way I see it, that either means that:

1. at least one of us is wrong about its necessity; or

2. its necessity is individual-dependent, so that we could both be correct and it is necessary for you but not for me.

I find the second one palatable but my secondary sources tell me that Kant was adamant that his a priori intuitions like the TA were not subjective. If that's correct then I think he'd roll in his grave at suggestion 2.

It is strange that earlier in this thread so many people point to England's gun laws as an example to be followed to reduce gun violence.

http://www.dailymail.co.uk/news/article-1223193/Culture-violence-Gun-crime-goes-89-decade.html

The statistics seem to be showing that there is a problem with those laws application as well. — Sir2u

Not being British you would not be aware of this. But quoting the Daily Mail as a source has about the same credibility as quoting a Trump tweet.

That is not to say that claims made in such sources are necessarily wrong - although they are more often than not. But if one even slightly suspects there may be a grain of truth in them, one has to go searching for credible sources that may corroborate the claim.

You have a knack for story-telling, getting people emotionally invested in your narrative. I'm feeling really cross about that clerk right now.

Cash in cash registers can be reconciled against sales. If he puts $5000 in the register without recording a sale on it, there will be a $5000 discrepancy, which will inculpate him as clearly as having it in his pocket did.

To escape, he would have to hide the cash somewhere that the police will not find it. The customer should not let him out of his sight, so that he can see where he hides it.

An eminently satisfactory joke. Well structured, good punch line. — Bitter Crank

Indeed. It made my day.

the customer has no proof that he paid for the diamond. — Michael Ossipoff

Didn't he get a receipt upon payment of the money? The OP does not mention whether he does, but only a fool would pay such an amount without immediately obtaining a receipt.

But even if the customer had been so unwise, they could immediately call the police and ask them to retrieve the $5000 cash from the clerk's pocket, dust it for fingerprints and ask the clerk to explain how they came to have $5000 cash in their pocket that had been handled by the customer. They'd be hard pressed to come up with a credible excuse, given the sign about the $5000 is sitting there in plain view, and the customer's testimony.

If you have given me $5,000 then I will give you the diamond. — Michael

I don't think that's a reasonable paraphrase of the sign. This version refers only to the present, and whether, at the time the reader is reading the sign, they have already given $5000.

The actual sign emphasises its difference from this paraphrase by use of the words 'at any particular time'. No matter how charitable one is seeking to be, one cannot ignore those words.

What if the words were removed? (goes back to OP to read sign while mentally eliding those words)
Without those words, I think the paraphrase might be considered realistic. However I think almost nobody would pay the money on the basis of such a sign. Those four words are crucial to tricking people into paying the money.

I wonder if any of the other Others in Perceptual Experience clicked that because, like me, they didn't know what any of the named terms meant.

I think the following incorporates your amendment:

$\forall t2\ \forall t1\Bigg(\bigg(Pays(C,5000,t1) \wedge (t1+1\gt t2 \gt t1)\bigg)\to OwnsDiamond(C,t1+1)\Bigg)$
where $C$ is the customer.

Substituting 1002 (10:02am) for $t2$, the 'any particular time', and 1001.5 for $t1$, the time the money was paid, gives

$Pays(c,5000,1001.5) \wedge (1002.5 \gt 1002 \gt 1001.5)\to OwnsDiamond(C,1002.5)$

We observe that the money is paid at 10:01:30 (ie 1001.5).
So both antecedents are true, so the consequent must be true, ie:

$OwnsDiamond(C,1002.5)$

But observation shows this is false. The customer does not own the diamond at 10:02:30. So the original statement must be false.

It still looks like the clerk was lying.

For one thing, you said that t2 equals or is greater than t1. But I'd said "...if, at that time, you have given $5000 to the sales-clerk..."

The sign explicitly specified a time after the payment was made.

Then you assign the same time value to t1 and t2.

That's just a first comment, from a look at the beginning of your argument.

For your argument to make enough sense to evaluate it, you'd have to change those parts of it. Only then would there be any point examining the rest of it. — Michael Ossipoff

I don't agree that those adjustments are necessary but, for the sake of furthering the discussion I'll accept them. Here's a version where $t2$ strictly exceeds $t1$. The money was paid at 10:01:30am.

Do you disagree with it? If so, with which bit?

$\forall t2\ \forall t1\Bigg(\bigg(Pays(C,5000,t1) \wedge (t2 \gt t1)\bigg)\to OwnsDiamond(C,t1+1)\Bigg)$
where $C$ is the customer.

Substituting 1002 (10:02am) for $t2$, the 'any particular time', and 1001.5 for $t1$, the time the money was paid, gives

$Pays(c,5000,1001.5) \wedge (1002 \gt 1001.5)\to OwnsDiamond(C,1002.5)$

We observe that the money is paid at 10:01:30 (ie 1001.5).
So both antecedents are true, so the consequent must be true, ie:

$OwnsDiamond(C,1002.5)$

But observation shows this is false. The customer does not own the diamond at 10:02:30. So the original statement must be false.

and then writing a long, elaborate argument in those term — Michael Ossipoff

The above post is much shorter than your statement of the problem in the OP!

If you disagree with it, with which bit do you disagree?

$\forall t2\ \forall t1\Bigg(\bigg(Pays(C,5000,t1) \wedge (t2 \ge t1)\bigg)\to GetsDiamond(C,t1+1)\Bigg)$
where $C$ is the customer.

Substituting 1002 (10:02am) for both $t2$, the 'any particular time', and $t1$, the time the money was paid, gives

$Pays(c,5000,1002) \wedge (1002 \ge 1002)\to GetsDiamond(C,1003)$

We observe that the money is paid at 10:02.
So both antecedents are true, so the consequent must be true, ie:

$GetsDiamond(C,1003)$

But observation shows this is false. So the original statement must be false.

Does anybody have a different formalisation to suggest?

Here is the sign:
"“If, at any particular time, you have given $5000 to the sales-clerk (under no circumstances will it be returned), then, within 60 seconds after your giving him that money, he will give you this diamond, and it will at that time become yours.”"

Call that assertion A1.

The 'at any particular time' is a universal quantifier so, under the rules of FOPL, it can be replaced by a reference to a specific time. Let's say the clerk says to the customer at 10:00am that the sign is true, and that the customer gives the money at 10:02. Under the rules of FOPL, if the sign was true at 10:00 then so was any version of it with the 'at any particular time' replaced by a specific time. So the following statement was true at 10:00*:

"“If, at 10:04, you have given $5000 to the sales-clerk (under no circumstances will it be returned), then, within 60 seconds after your giving him that money, he will give you this diamond, and it will at that time become yours.”"

To prove this, call that more specific assertion A2. Under the rules of FOPL we have A1 --> A2, which is a tautology and hence true at any time at all. Call that tautology T1. Adopt a conditional hypothesis that A1 was true at 10:00. Then by Modus Ponens on A1 and T1 we deduce that A2 was also true at 10:00am.

But A2 is false at any time, because money was given at time 10:02 and a diamond was not given by 10:03. So in particular A2 was false at 10:00. By contradiction that entails that our Conditional Hypothesis that A1 was true at 10:00am must be false.

Hence A1 was false at 10:00am when the clerk asserted it was true. So the clerk lied.

Let's not discuss that any more.

if two parallel lines cross a line segment, they have equal angles — Metaphysician Undercover

That is a claim, not a definition. Observe how it does not say 'we define X to mean' or any of the equivalent forms of words that flag a definition.

The claim is only true if we adopt the parallel postulate or something equivalent to it.

No, the parallel lines never meet, it is impossible, because the definition indicates this. If they meet they are not parallel. The point being that we must accept the definitions and adhere to them whether or not there is any such thing as infinite lines, or parallel lines in the physical world — Metaphysician Undercover

The parallel postulate does not define parallel lines. They are defined in Book I Definition 23, as being two lines that never meet.
What the parallel postulate does is assert that two lines that cross either end of a line segment at non-right angles are not parallel. It doesn't actually say 'not parallel' but rather gives a property that is equivalent to being not parallel.

So the postulate neither defines parallel lines, nor asserts that there are any. I presume the existence of at least one pair of parallel lines must be a theorem that is deduced from that and other postulates. Although, as I said much earlier, I believe that statement of the postulate is incomplete, and needs the words 'and only on that side' to be added. Otherwise the postulate does not exclude elliptic geometries, where all non-coincident pairs of lines meet in two places.

So my question to you is: do you think my inference that 'what is causing the interference pattern is outside, or not a function of, space-time' is indeed 'gobbledygook'? Or do you think it's a valid inference? — Wayfarer

The problem is that by asking what 'caused' something you are moving from physics to philosophy - metaphysics, and there's quite a bit of sensitivity on physicsforums about discussions veering off into philosophy. They try - not always successfully - to maintain a clear boundary between physics and philosophy and it looks like you inadvertently crossed it. I thought the reaction was a bit harsh, especially as you, not being a regular there, would have no reason to be aware of that sensitivity.

I wouldn't comment on the validity of your inference because I see discussion of what caused what as frustratingly ambiguous. But I can observe that increasing the time between photon emissions also increases the average distance between photons, which reduces the frequencies of interactions and hence the extent to which interaction effects distort the pure proportionality discussed in the A to the second Q. And that maybe sounds something like what you were saying.

So, say two lines met a billion parsecs away. In that case do they start out truly parallel? If they do, then at what point do they cease to be parallel?

They always look parallel, and that's what matters to our intuitions. There is no part of the triangle we can look at in which the bit we can visualise doesn't either look like two parallel lines, or one line when they are so close together that we cannot distinguish them.

So if we want an intuitive parallel postulate, I imagine it would have to be something like:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to an amount that is VISUALISABLY less than two right angles, then there is some VISUALISABLE distance such that, if the two lines are extended for that distance in the direction on which the angles sum to less than two right angles, they meet.

and this postulate would be met by any space that is no more than very slightly curved, which would include our real world space.

The parallel postulate does not say what, based on your post, you appear to think it says.

What struck me about the fact of 'rate independence' is that it shows the idea of 'the wave' is actually a metaphor - it's not as if the probability wave is an actual wave, because it is not defined by time.

Yes, it's a bit like if we snap-froze the ocean while it was wavy. Then we'd have a physical wave shape, but one that was constant across time.

I think one source of the confusion that often arises here is that we can predict the interference pattern using classical wave theory, ie avoiding QM altogether. In that case we are using actual electromagnetic waves emanating through the two slits. Then at each point on the screen the wave manifests as complementary sinusoidal variations in electric and magnetic potential over time. The result of the two waves interfering is that the amplitude of those variations on the screen varies as we move across the screen, giving high amplitude in the middle of the light bars and low amplitude in the dark bars. In that analysis the EM waves are moving in both time and space. But the static 'probability wave' of QM plays no part in this calculation, even though they give the same results.

It is often forgotten that the double-slit experiment is entirely explainable using classical electromagnetic wave theory. QM is only needed to explain various tricky variations of the experiment, such as the one you describe where photons may appear at the screen a rate of only one per second.

What seems to be the challenging philosophical issue is, however, the ontological status of the probability field — Wayfarer

Indeed. I approach this by regarding 'laws of nature' as descriptive rather than prescriptive. That way they do not need ontological classification. They are a just a tool we use to visualise what is going on and to make predictions. I accept that many, possibly most, people find that unsatisfying.

If one wanted to attribute an ontological status to the probability field, what do you think of regarding it as some sort of Kantian noumenon? We can never observe the probability field itself, only phenomena that arise from it (photon strikes). That sounds a bit noumenish to me, but I'm a bit of a noumenon newbie (just had to find a way to get that phrase in there).

The average direction across all galaxies is away from one another. However that does not mean that every single pair has to be moving apart. Each galaxy has its own idiosyncratic motion, called its 'peculiar velocity', which is measured as deviation from the velocity it would have if it followed the average of all galaxies. Two galaxies near one another can have peculiar velocities towards one another giving a convergence speed that is bigger than the divergence that would arise if they both followed the overall average, so they collide.

Something that might help is to consider a child's helium balloon that she has lost hold of and is floating up into the sky. As it rises, the outside pressure reduces so the balloon expands (eventually it will burst!) so the average motion of helium atoms in the balloon is away from one another. But individual helium atoms in the balloon have big peculiar velocities and are still banging into one another like anything - just like some galaxies.

'The independence of an interference pattern produced by a given number of photons from the time required for those photons to be registered is simply further confirmation of the assertion that each "interferes with itself" ' - which personally I find an absurd explanation, even though it is one of the expressions that is commonly used.

I agree with you. I don't find the phrase 'interferes with itself' meaningful or helpful.

My current way of looking at it (it has changed in the past, and likely will again in the future) is that the wave part is a probability field for a photon striking the screen. With one slit, that probability field is strongest in the centre of the screen and decreases as we move away from there. With two slits, the field strength has an undulating profile as we move across the screen, with peaks and troughs. We call the peaks 'bars'.

If we dim the light, the probabilities decrease, but the fields retain their shape - single blotch or series of bars according to whether there is one or two slits. The blotch or bars are just weaker.

In any given small region Photon strikes appear by a random process (a Poisson process) whose frequency parameter ('probability') is the average field strength in that region. This builds up the pattern over time - quickly if the light is strong and slowly if it is weak ('slow photon rate'). There is no expected difference between the pattern built up from a constant light source over period of length T and that built up from a light source one millionth as strong over a period of one million times T.

This explanation satisfies me, without having to introduce any notion of photons interfering with one another, let alone with themselves. The interference is between the probability fields of the two slits - the 'waves' as some might put it.

I have also been thinking about what it would mean to have an intuition of the parallel postulate, if that means being able to visualise constructions that demonstrate it.

Consider a line segment AB of length 1cm, with a line L1 going through point A at right angles to AB and another line L2 going through point B at right angles. The parallel postulate says that L1 and L2 never meet.

Now replace L2 by a line L3 through point B, that is at an angle that differs from 90 degrees by such a tiny amount that it intersects L1 at point C, a distance of a billion megaparsecs from AB.

Can you imagine triangle ABC? I can't. If we look at the AB end, what we see looks like one end of a rectangle. If we look at the C end, what we see looks like a single straight line. I cannot hold the whole triangle in my mind's eye.

The parallel postulate says that L2 meets L1 but L3 does not. But I cannot distinguish in my mind's eye between AB with L1 and L2 attached and AB with L1 and L3 attached.

So, for me, the parallel postulate is not something that can be visualised.

Another way of saying that is that 'infinity is a very long way'. It is such a long way that the difference between 'these two lines will never meet' and 'these two lines will meet at a point a billion megaparsecs away' is meaningless - to me at least. Of course I can do calculations with it that have different consequences for the two cases. But that is not visualising it, and I suggest it is not intuiting it either.

So I submit, your honour, that the parallel postulate is not intuitive.

I've had another response to my question on measuring curvature of space. It makes the excellent point - which I had completely missed - that the path traced out by a laser beam in a constant-time spatial hypersurface is not necessarily a geodesic (straight line) of that hypersurface. Certainly there is no obvious mathematical reason why it should be so, even though our instincts expect it to be. Unless that traced-out path is a geodesic, even the experiment involving the three space stations and lasers may be unable to directly demonstrate a spatial curvature. One would need to do a very complex calculation that decomposed the deviation from 180 degrees of the triangle's angle sum into a component attributable to curvature and a component attributable to the laser paths not being geodesics.

So curvature of space, if it exists at all (recall the observation above that there exist coordinate systems (reference frames) within which entire regions of space are flat), cannot be directly observed even with extremely high tech equipment. One needs to be proficient in GR, and very patient and determined and have a lot of time on one's hands, even to do the calculations that might indicate a curvature.

I suggest that, if curvature of space cannot be directly observed, but only inferred from long, complex calculations that most people would not understand, it interferes with or invalidates our intuition of space in the TA not one whit.

I believe the statement was not true when the person first inquired, because the words 'at any particular time' are not constrained to cover only the past, so they cover the future too. It is a universal quantifier: for all t.

If the person is inquiring at time t1, the quantified part of the statement is true for values of t less than t1, by virtue of the above truth tables Null implication) but it is not true for values of t more than or equal to t1. Hence the statement is not true at time t1 because it is universally quantified and it is not true for all values of t.

One might think that removing the universal quantifier will thereby render it true. But then the 't' in the statement (implied in '60 seconds after you have given him your money') becomes a free variable, and in FOPL, a formula containing a free variable entails the version of the formula in which that variable is universally quantified (rule of universal quantification), so we are back where we started.

So I don't think it works. The store clerk lied.

Now show me that you can imagine intrinsic curvature in the same way. — Agustino

I can only repeat what I said above, that we don't need to imagine it. Cognising space as a Riemannian Manifold is not non-Euclidean, but aEuclidean (think of the difference between immoral and amoral). It is uncommitted as to whether the space may be curved, as long as it is not heavily curved.

I would call the experiments I described a way of 'imagining' a non-Euclidean 3D space. But I feel no need to argue if you don't consider that imagining.

Suppose the curvature is very high, such that if you take 10 steps in one direction, you return to the same point where you started. This is a thought experiment, an unrealistic one, but it's useful. Suppose there are a series of poles, 1 step apart, in front of you, with the pole right next to you being red (so that you can keep track of when you return), while the others are some other colors. How would this visually look to you? — Agustino

Say there are three poles, coloured red, yellow, blue, at distances of 3 1/3 steps away from one another, in a straight line in the direction I'm looking. There can't be more than three because the fourth pole would be where the first one is.

As I look along the line I see an infinite series of poles: red, yellow, blue, red, yellow, blue, etc. Next to every red pole is an image of me, seen from the back. The images of poles and of me diminish in size as they move along the line of vision, just as a series of poles beside a long, straight road does.

It would be somewhat similar to what one gets when one stands between two opposing mirrors, except that the view of myself would always be from behind. You may be interested in this essay I wrote about something like this - what happens when we point a TV camera at its monitor, inspired by a comment Alan Watts made in one of his talks. There are some pictures and videos in it that I find quite cool.

That's true; they would not be straight in the vertical plane, because they would curve to remain parallel to the curvature of the Earth. What if we could build a rail line into space; it could be straight and parallel in both planes I think. — Janus

If the rail line were stationary relative to Earth, the lines could not be both straight and parallel, because in that reference frame the spatial slices are curved. Since parallelness is necessary in order for the train to be able to run but straightness is not (trains can go around curves), we would have to give up straightness, rather than parallelness.

It may be useful to be clear what we mean by parallel. What I mean is that if we draw a straight line perpendicular to one track then it meets the other track at right angles.

Interestingly, if the track were in free fall towards Earth then it may be possible for the lines to be both straight and parallel. That's because, subject to a few other initial conditions being met, its reference frame could be the one I referred to earlier as one in which curved spacetime can have flat spatial slices. It would make the devil of a mess when it hit the Earth though.

How do you imagine a 3D, non-flat space? How do you imagine intrinsic curvature? Hopefully, you won't say that you do via analogy to extrinsic curvature. — Agustino

I'll describe below how I imagine it, but that's beside my point, which is that I think we don't need to imagine it. I think all we need to cognise the world is the bolded list of items in my previous post, and we get that from any Riemannian Manifold, whether flat or curved. When we use those things in navigating the world we can remain uncommitted as to whether the space is slightly curved or perfectly flat.

Now to reply to your specific question. You are right, it is weird to imagine. Here's a couple of ways:

1. Two spaceships set off on a journey, travelling initially parallel and starting 1km apart, going at the same speed and steering straight ahead. If the space is flat they will remain 1km apart. If not, they will subsequently measure that they are getting further apart if the space is hyperbolic, or closer if it is elliptic.

3. Set up three space stations 1, 2, 3 in deep space, each firing a laser beam at the next: 1 to 2, 2 to 3, 3 to 1. Each measures the angles between the incoming and outgoing beam. The stations are floating freely, not firing rockets to accelerate. The three angles will add to 180 degrees if the space is flat, less than that if hyperbolic and more than that if elliptic.

We can build rail lines extending thousands of kilometers and the rails are (not perfectly, but on average, parallel). — Janus

This got me thinking. How would we build a rail line to circumnavigate the equator, if there were a 5m wide land bridge all the way that followed the great circle of the equator? Say the land bridge is perfectly level (constant altitude above mean sea level) and extends at least 2.5m to either side of the equator at every point.

I'm pretty sure that the answer is that the rails would always be parallel and equidistant, but what we'd have to give up is the requirement that they be 'straight' - what's called a 'geodesic' in tech terminology. Say the gauge is standard and the centre of each rail is always 717.5mm away from the equator - one in the Northern and one in the Southern hemisphere. Then neither rail can follow a great circle but instead is constantly curving away from the equator at an incredibly small, constant rate.

So the lines would be parallel and a constant distance apart, but they would not be perfectly 'straight'. However a train could run along them with no difficulty at all.

Why can't the rails both be straight? Because a straight line on the surface of a sphere is a great circle, and any two great circles will intersect at two antipodal points. It would however be possible to make one of the rails a great circle and the other one not - eg if one rail followed the equator and the other were in the Southern hemisphere..

(1) What ordinary people mean by intuition cannot be used to defend Kant, who uses that word differently, and thus means different things by it than ordinary people. — Agustino

My intention is to defend not Kant, with whom I disagree on many important things (although I do have enormous admiration for him), but what I see as the amazing insight and usefulness of his notion of the Transcendental Aesthetic (TA). In the discussion over whether you and I find non-Euclidean geometries unintuitive, I see that as just a reflection on your and my particular cognitive processes, rather than about the TA, which is suggested to be universal to autonomous humans.

My interpretation of the TA, which has evolved in the course of this discussion (thank you everybody - this forum can be such a learning experience), is that humans process sensory input in a framework consisting of two Riemannian manifolds: a 3D one that we call 'space' and a 1D one that we call 'time'. That Kant did not describe it this way I ascribe to the fact that the language necessary to express that did not exist in his time.

Space as a 3D Riemannian manifold gives us points, lines, shapes, volumes, angles, directions, relative positions, insides and outsides, and distance.

As I see it, that, together with time, is enough for us to navigate, imagine and discuss the world. At most I would add a requirement that any curvature not be too extreme, because if that were the case we might find ourselves back where we started if we walked one metre (if the space were elliptic), That requirement is completely consistent with the region of the universe in which we evolved, and which we now inhabit.

It may well be the case that for some people the space manifold is also perfectly flat (ie no curvature, not even if unmeasurably small), as you report to be the case for you. But I suspect that is an individual variation, rather than a universal feature. For my own case, It is not necessary in order to obtain all the concepts listed in bold text above.

Which way? — Agustino

In a way that does not require the space manifold I use to be perfectly flat.

For Kant intuition means something closer to perception — Agustino

Yes, I mean what ordinary people mean by intuition, not what Kant means . He uses words too weirdly for me.

is non-Euclidean geometry perceptible in your mind's eye? — Agustino

Yes. It may be, as you say, cos I'm a mathematician. Or maybe I'm a mathematician cos I look at things that way.

So this new reconceptualisation was not a pure intuition as per Kant's definition of the term? — Agustino

I think the concepts are a lot easier than the axiomatisation. The concepts are intuitive (again, maybe only to me), but the axioms are not.

For some reason revealing of my own unacceptable prejudice I'm slightly offended that you think I'm American. Anyway, I'm English, Our biggest problem is the BNP, UKIP etc., but the problem of Trump I see as an example, not an exception, and the British response has been instrumental. We've basically said that we don't want him over here to speak, that nothing he's got to say is of any interest to us. I think that's a very powerful expression of the contempt in which we hold his views, much more powerful than letting him over here and debating them, as if they had any kind of legitimate reasons that might require some thought. — Pseudonym

I think that's exactly the correct response in that case. I hope it lasts. Do you think May will give in and invite him over at some stage, despite the unpopularity of such a move with the British people?

For the first part of your post - I think your assessment of our respective positions is correct. We seem to agree on aims, but disagree on methods - at least as far as local talks and demonstrations go. And we seem unlikely to persuade one another. I hope you turn out to be right and I turn out to be wrong, because that will mean that your 'de-platforming' efforts have been successful in diminishing the influence of the UKIPs, BNPs and Milo Yiannopoulos's of this world..