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A Hard Problem of Grammar: A formal logic analogue to Nagel’s bat

The hard problem of consciousness, and variations on it, revolves around the difficulty of explaining mental phenomena—I see and smell a rose; I think about my work; I feel pleased by good news—in materialist or physicalist terms.

The presumptive barrier that prevents neurophysiological research answering this question, is that objective observations which can be made by a researcher — such as, electrical current moves through these neurons; a biochemical cascade releases such and such a hormone — appear to be about completely different things than the subjective experiences of a conscious experiencer.

This objective/subjective gap has also been called a first person / third person gap: how can third person sentences such as “that neuron spiked” possibly relate to a first person sentence such as, “I see red.”

Expressed this way, the problem can be studied with formal languages. This appears to make it provably insoluble. It’s a hard problem of grammar: there is no sound deduction from a set of 3rd person sentences to a 1st person sentence in any formal logic.

  • If subjective experience could be explained in objective terms, then that explanation—if it is a rational one—must be expressible in formal language.

(This claim may not be obvious. It’s like claiming that some form of the Church-Turing thesis applies not only to mathematics and logic, but to rational discourse more widely, including empirical research.  I’m suggesting that any argument which is genuinely rational, can be expressed in a formal language. If some part of the argument can’t be formalised then I think we will discover, on inspection, that it’s because the argument isn’t rational. Either it will be a non-sequitor, or it will be an appeal to emotion, or an ad hominem attack, or somesuch).

  • If so, the formal version of that explanation would have to, at some point, define the word ‘I’ in ‘3rd person’ terms, that is, without relying on any first person noun or verb or other part of grammar.

(This is essentially an assertion that some form of the Craig interpolation theorem can be proven for any formal grammar suitable for rational discourse. The Craig interpolation theorem says (more-or-less), that given a set of sentences only about apples, you cannot validly deduce from them a sentence about oranges. I suggest that any formal language that does not satisfy this constraint is not a language we can use for rational discourse. It may still be fine for poetry; but not for being rational).

  • I assert that any such attempted definition will, on inspection, turn out to be invalid.  That is, when we look at any such purported definition, we’ll be able to see that it doesn’t quite work and hence that the explanation which it supports will also fail. I do of course look forward to being proven wrong, but there aren’t any attempts on the table so far.

I think any attempt to define 1st & 2nd  person words – I, you, me, we  – in 3rd person terms fails. Any definition using only 3rd person terms can only succeed in defining 3rd person ‘things’.

The thought is somewhat similar to Chalmer’s “Structure and functions” argument: anything you can define with structure and function will itself be structure and function. I think the grammatical argument is stronger, surprisingly (one doesn’t expect arguing about grammar to prove anything!), because 1st and 2nd person speech and relationships is, and always has been, a core part of the reality of human experience.

Intuitively, throughout the modern era, people have always felt that a reductionist materialist account of humanity surely misses something. The grammar of every human language (at least, every one that I know of) embodies that fact.

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Mathematics: a Definition

I propose a definition of mathematics:

Mathematics is the deductive analysis of structures

  • Deductive, because empirical data does not generate mathematical results; only logical deduction does so.
  • Analysis, because mathematicians tease out the consequences of the definitions of structures, rather than merely admiring, or using, or playing with them.
  • Structures because … well, this is the question: what do mathematicians study?

What do Mathematicians Study?

The OED definition of mathematics relies on a rather post hoc list, as the “etc” acknowledges:

“The abstract deductive science of space, number, quantity, and arrangement, including geometry, arithmetic, algebra, etc., studied in its own right (more fully pure mathematics), or as applied to various branches of physics and other sciences (more fully applied mathematics).

Shorter OED 2007

Without the etc, this would need revision every time a new area of mathematics opens up. It is more like a rough description than a definition.

But every branch of mathematics straightforwardly has this in common : it analyses a particular structure (or a family of structures; which is still a structure), and deductively analyses it, that is to say draws out its properties and relationships to other structures.

There are favoured structures. Numbers, of course. Then the Euclidean plane, which is the structure of lines and points on a flat surface. These favoured structures define the familiar major areas of mathematics–number theory, algebra, geometry, analysis. “Progress” in mathematics divides into, on the one hand, discovering new things about known structures; and on the other hand choosing new structures to study.

New structures may be chosen for the light they shed on old ones: complex numbers, for instance, shone a new light on algebra, as did topology on geometry. Formal logic & computer science intended to shine light on the very processes of mathematics itself. Whenever a new structure is found to have interesting properties it may become a part of mathematics and even, if there is work enough in exploring it, be dubbed a branch of mathematics, which is perhaps the ultimate mathematical status.

The virtue of naming structure as the subject of mathematics is that it becomes easy to say whether something is or is not mathematics: anywhere there is a structure that can be analysed deductively, there is a subject of mathematics. The ad hoc element of the definition is banished.

It also reminds us not to be surprised every time a new branch of mathematics opens up. If it moves, or even if it doesn’t, it’s fair game to a mathematician.


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Rule #1 of Coffee Club : Keep It Fresh

“Think of when you cut into a banana, how quickly that slice of banana oxidizes. Now think of how big that slice of banana is compared to these tiiinnnyyy granules of espresso.”

“They oxidise quickly! Get rid of them, grind them fresh!”

A well-packed 23 minutes from US Barista Champion Heather Perry on making good espresso:

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Deep Learning & Unintended Algorithm Bias

This was a 5 minute talk on deep learning for the very excellent @chesterdevs. Like others talking about deep learning, I took visuals and the face-learning example from the landmark 2012 paper, Quoc Le/Google/Andrew Ng paper, “Building High-level Features Using Large Scale Unsupervised Learning.”

Only afterwards did I notice that the subset of images which their system show as “most like a face” from their test set were 90% male and 90% white, as is the prototypical face that the machine outputs.

And so we have a neat demonstration of unintended algorithm bias: their input was 10 million randomly-chosen youtube videos; the output was white and male. I bet they didn’t expect that.

A salutary reminder that—as the hard-working statistician will tell you—“random selection” does not mean “unbiased”.

Colorless Green Ideas

It can only be the thought of verdure to come, which prompts us in the autumn to buy these dormant white lumps of vegetable matter covered by a brown papery skin, and lovingly to plant them and care for them. It is a marvel to me that under this cover they are laboring unseen at such a rate within to give us the sudden awesome beauty of spring flowering bulbs. While winter reigns the earth reposes but these colorless green ideas sleep furiously.

C. M. Street

https://en.wikipedia.org/wiki/Colorless_green_ideas_sleep_furiously#Attempts_at_meaningful_interpretations

Science & Logical Positivism

New Scientist once published a ½ page letter in which a working scientist ranted that philosophy was all meaningless and the Only Worthwhile—and Obviously True—philosophy is Logical Positivism.

But logical positivism is distinguished amongst all philosophies as the one which disproves itself in a 2-line proof.

Logical Positivism Premise #1 : All meaningful statements are either analytic (that is to say, statements of mathematics or logic or some other tautology) or else statements of empirical fact, and any sentence that is not in one of these two categories is strictly and literally meaningless.

2. If premise #1 —which is not a tautology, nor a statement of mathematics or logic, nor a statement of empirical fact—is true, then by premise #1, premise #1 is itself strictly and literally meaningless, and therefore cannot be true.

Customise Macos XQuartz : xinitrc doesn’t work

If you installed XQuartz and are, for instance, irritated by the small white xterm window you get, you might try customising it in the usual way by editting an .xinitrc file. If only.

Instead, try this:

defaults read org.macosforge.xquartz.X11

to see all the settings; or to permanently change the startup xterm window, something like:

defaults write org.macosforge.xquartz.X11 app_to_run \
 "/opt/X11/bin/xterm -fa Monaco -fs 12 -fg green -bg black -sb -sl 1000"

Or, if you have installed a better bash with homebrew, then e.g. :

defaults write org.macosforge.xquartz.X11 app_to_run \
  "/opt/X11/bin/xterm -fa Monaco -fs 12 -fg green -bg black -sb -sl 1000 -ls /usr/local/bin/bash"

You can check your syntax before writing the default just by running your quoted command in a terminal, and then watch as XQuartz opens and xterm runs your shell:

~/Source/Repos/VMs] /opt/X11/bin/xterm -fa Monaco -fs 12 -fg green \
    -bg black -sb -sl 1000 -ls /usr/local/bin/bash

To set the default for a new xterm window from the XQuartz Application menu, the menu itself lets you edit the command.

In short, read the FAQ : https://www.xquartz.org/FAQs.html.