Triumph and Biumph

For some reason, this went through my insomniac mind. "Quadraumph"?

My best equations

I hope I am remembered for these:

The 'dot' in the first expression is function composition. The '*' in the second is inbalance.

The first says that 'e ^' estimates a classifier 'e' by measuring (mu), picking a best model with those measurements (phi) and classifying that. In other words, things are classified by how the nearest ideal example is classified - stratified by measurement values. I.E. recognition is a lifting.

The second says that conflict is resolved. 

Plum cake

 I had too many ripe plums I did not want to eat. So I sliced them up, salted them, baked them at 200 for an hour and then added them to a standard white cake recipe - which I cribbed from some other plum cake recipe:

1 stick salted butter

3/4 cup sugar

1/2 tsp salt

2 eggs

1 tsp baking powder

1 cup flower

Cream the butter and sugar, stir in eggs one by one, mix as fluffy as possible. Fold in dry ingredients, mixing as little as possible. Add baked plum fragments. Bake 350 for ~1 hour (or until fork comes out dry).

Note there is a lot of liquid in the plums. Pre-baking the plums removes some of it. Baking the cake a little longer than usual is needed. The source of my cake recipe says it is good to let the juices diffuse into the cake overnight before eating it.

Not minding that it hurts

The famous scene in Lawrence of Arabia, where Peter O'Toole burns himself with a match. His colleague asks: "What is the trick?". O'Toole answers: "The trick, William Potter, is not minding that it hurts".

This seems like pretty good advice for dealing with something that makes you sad.

Update: but it is awful advice for someone who could get an infection.

A guilty little secret of formal logic

The problem with borrowing words from natural language (like "and", "or", "if") to define entities of formal logic, is that there is nothing to prevent the words from being used in their original natural sense, in the middle of what is  supposed to be a formal argument. 

The reason this is not actually a problem is because the formal definitions are close to being correct and a good duplication of the natural word usages. I think it is possible to define an exact duplication of the natural word usage - so that we do not need to worry about that any more.

How do you know when Bertrand Russell has completed a proof?

 Because he stops talking.

Woods Hole becomes Mathematics

I want to tell anyone who finds themselves reading this, that I have benefitted greatly from living in Woods Hole (or being able to stay here in the summer and now all the time). I sat here, year after year, peacefully thinking about the most abstract things possible. And now, about 67 years in, I am writing my final ideas in the paper The Moving Topic. I look out the window: it is still peaceful out there.

Please consider, that The Moving Topic to be the distillation of Woods Hole into logic. I think the place gets the credit.

Overheard at the grocery store

Living the dream?

One day at a time.

The definition of Math

I have decided that the most characteristic thing about Math is the way it ignores intermediate steps. I call that the original sin of mathematics. Every equation, like "2+2" equals "4" is an ignoring of the difference between two narratives. They end up in the same place. 

Consider the Piaget concept of "object permanence", where a child learns that things can be hidden but still exist. Isn't that lesson based on knowing the outcome is the same, whether or not the object is hidden? I take it then, that this early cognitive processing is, in some sense, Math. Even earlier a baby would have learned about object "constancy", meaning moving things did not change.

Why formal logic definitions are a mess

I have a growing list of complaints. One basic problem with logic is that it tries to be timeless and tries to have the meaning of an expression limited to the words in the expression. Narrative reaches into the past and the future with it's implicit meanings. Here is a list of problems with the definitions of formal logic.

One basic problem is that math co-opts natural language but has no safeguards to ensure that the co-opted version is what people actually use in the midst of a proof. Eg phrase "if A then B" subsumes so many different concepts that the transitivity of the relation is in question, having only been established for the things being subsumed but never for a mixture of such things. Again, there are no guardrails to ensure this. [I actually think Russell was aware of this but, other than admonishing the audience, he proceeds.]

Truth: 

• This is called an 'undefined' constant. It is never established that way. In fact it has a several distinct meanings that are being suppressed for the definition but (inevitably) used in actual practice. There is no way to establish the truth of a proposition using the rules of logic, so what is the point of relying on 'truth' to define logic?
• Put another way, to say "true" is an undefined constant leaves it unclear how we know something is true? For example: "If you count from one to ten then you must pass five". This is true but how can we believe it is the same as the  undefined constant version of "true"? In fact, why would it be? 
  

Circularity: 

• I count three places in the definition of "and" where they use the 'and' concept. Once explicitly, to define 'and' using "and"; and then subtly in the 'newline' needed for additional rules of "detachment". Alternatively you see a definition in terms of truth tables which is un-objectionable but also empty. Also boolean tables have rows and columns which serve as 'and' mechanisms. Those un-acknowledged  mechanism are what actually performs the definition, not the undefined zeros and ones.
• Consider this bit of gibberish:

Logical implication is a logical relation between two propositions in which the second is a logical consequence of the first1234. It means that if the first proposition is true, then the second proposition must also be true2534. Logical implication is also known as implication, logical consequence, implies, or If... then234.

So who defines "logical consequence"? And what is with the "if...then" in the second definitions? 

Overlapped Definitions:

• The idea that there are more logical operators than logical operations shows that the latter have not been factored correctly.
• Isn't it a little embarrassing, every time you say "or" to follow it with "but not both". The word "or" in natural language means choose one and has never meant "both". So, OR and XOR. Which is it? Also, why would natural language speakers find themselves using a locution like "and/or"?
  

Deliberate Obfuscation:

• The meaning of "if A then B" is entirely artificial, co-opting natural language and then adding insult to injury by bullying the reader. It means: replace A with B cuz I said so! To be fair, it is a shorthand for skipping steps and its transitivity is established by ignoring intermediate steps.

Lack of clarity on 'universal' vs 'particular'

• It is elementary in computer programming to distinguish between a class definition and a class instance. The idea is less crisp in textbooks on logic. Mathematicians submit themselves to obfuscating different kinds of sets, so "man", "men", "mankind", "Socrates", ... these are difficult for ma-man Russell.

Non-atomic "atoms":

• The "universal quantifiers" 'for all' and 'there exist' are not fundamental but derived from simpler operations: stepping through elements of a set, testing for each element for pattern match/mis-match, employing different exit strategies for ending the testing. Iteration and pattern matching are clearly more fundamental. So why are these derived concepts taken as starting points? [Answer: so that the ideas of "all" and "some" can apply when we skip the step of defining the iteration/matching/exit.]

Finally, there are serious issues with treating all propositions and entities as equivalent and existing on the same "playing field". Bertrand Russell was the first to express doubts about the types of entities collected in a set but logicians, concerned with the infinite, overlook problems with very finite entities. No one would say, in natural language, that the a dot's color is "red and green". It would be considered incoherent for someone to say a cup was "red or shiny". This un-secured nonsense shows the necessity of treating different types of attribute in different ways.

Update:  There is a different thing wrong with the example of "All men are mortal". You do not need the "All" and the issues it brings with it. Instead you can simply observe that the definition of 'man' adds requirements to the definition of 'mortal'. You do not need any single example for the a priori truth of the statement. I am not sure what Russell's excuse would be for getting this wrong. Another place where he is hoisted on the petard of his own intention/extension ambiguity.

Hollyhocks outside my window

 

Inventing little languages

For the record, in college I was worried about Ontology and the name relation. I studied the inverse of quotation marks, as a notation to help express ideas about naming. I played with definitions of 'truth' leading to ideas of probability in finite collections of 'if-then' statements. I majored in Philosophy. Took symbolic logic (from a guy named Webb) and was puzzled by the idea that "but" meant the same thing as "and".

For the record, I spent a while in early grad school thinking about simplified algebraic entities I called "lingos". They were sets, closed under a binary operation. I played with different forms of set theory being embodied in different forms of element '∈' definitions. I was always interested in little algebraic systems. I played with semi-groups, and normed semi-groups. None of it was very deep. I had to quit all that to focus on passing Math exams and get a Ph.D. in Mathematics.

Later as a software engineer I spent a good deal of time designing user interface features and time trying to understand basic programming constructs such as memory versus program - data versus instruction.

And finally, I have been looking hard at the basic meanings of "and" and "or" ever since I read Bertrand Russell admitting confusion about them (which he forgot about later in his career). That might have been before college. Understanding these has only been possible lately with a clear articulation of the '*' and ','  narrative elements.

The point is that I have actually been thinking about these things my whole life.

What happened to the 4th of July?

It's the 4th of July and 3:20 PM and I have not yet heard one single explosion. Used to be the bangs would have started the night before. What is going on? I cannot talk to my neighbors cuz they might turn out to be Zionists.