Tuesday, September 22, 2026
Tuesday, September 15, 2026
Friday, September 11, 2026
Thursday, September 10, 2026
Tuesday, September 8, 2026
Thursday, September 3, 2026
Tuesday, September 1, 2026
Cause and effect can betray even the smartest amongst us!
When is C a cause of E?
Take this simple example: Pluto's barking causes Dick to wake up.
Take this simple example: Pluto's barking causes Dick to wake up.
We need four steps:
1. The cause must precede the effect,
So, we have:
2. It's nearly impossible for the cause to happen and the effect not to happen. Another way of saying it: if the cause had not happened, the effect would not have happened.
3. The cause makes a difference (without the cause, there's no effect),
4. There's no common cause. Think of a person suffering from depression. You could frame their depression using the common-cause relationship as well. Depression leads to a lack of motivation AND a lack of appetite.
4. There's no common cause. Think of a person suffering from depression. You could frame their depression using the common-cause relationship as well. Depression leads to a lack of motivation AND a lack of appetite.
So, we have:
1. The barking precedes Dick's waking up,
2. It's nearly impossible for Pluto to bark and Dick not to wake up,
2. It's nearly impossible for Pluto to bark and Dick not to wake up,
3. Without the barking, there's no waking up,
4. Suppose both the barking and waking up are caused by a person jumping Dick's fence and making a noise which causes Pluto to bark and Dick to wake up.
Necessary, sufficient, and contributory causes:
Pluto's barking is a necessary cause of Dick's waking up if the barking precedes the waking, but it doesn't imply the latter will occur, i.e., Dick may not wake up (it's not guaranteed).
Pluto's barking is a sufficient cause of Dick's waking up if the the barking necessarily imply the waking, although another cause may also contribute to Dick's waking (noise coming from the bathroom where Dick's wife was taking a shower). Here the presence of the waking doesn't guarantee the prior occurrence of the barking.
A cause is a contributory cause if it's one among several co-occurrent causes. In general, there is no implication that a contributory cause is necessary, though it may be so.
Reciprocally, a contributory cause is not sufficient for the effect, because it is by definition accompanied by other causes, which would not count as causes if they are sufficient, for your information, most causes are contributory, i.e., there are more than one.
4. Suppose both the barking and waking up are caused by a person jumping Dick's fence and making a noise which causes Pluto to bark and Dick to wake up.
Necessary, sufficient, and contributory causes:
Pluto's barking is a necessary cause of Dick's waking up if the barking precedes the waking, but it doesn't imply the latter will occur, i.e., Dick may not wake up (it's not guaranteed).
Pluto's barking is a sufficient cause of Dick's waking up if the the barking necessarily imply the waking, although another cause may also contribute to Dick's waking (noise coming from the bathroom where Dick's wife was taking a shower). Here the presence of the waking doesn't guarantee the prior occurrence of the barking.
A cause is a contributory cause if it's one among several co-occurrent causes. In general, there is no implication that a contributory cause is necessary, though it may be so.
Reciprocally, a contributory cause is not sufficient for the effect, because it is by definition accompanied by other causes, which would not count as causes if they are sufficient, for your information, most causes are contributory, i.e., there are more than one.
In distinguishing the strength of evidence for causation from causation itself one gets:
“Probably causal” & “potentially causal”. They describe how confident we should be from the evidence given.
Potentially causal: There is a plausible causal connection, but the information given is insufficient to establish that C made a difference to E.
Probably causal: The evidence described gives us strong grounds for thinking that C did make a difference to E, even though causation is rarely demonstrated with absolute certainty in ordinary cases.
Here is the spectrum:
mere coincidence → possible causal relationship → probable causal relationship → well-established causal relationship
The counterfactual is what moves us along that spectrum.
Suppose: Maria drinks coffee → Maria can't sleep.
We initially have only temporal succession.
Then we discover: On evenings when Maria doesn't drink coffee, she sleeps normally.
The counterfactual becomes much stronger: Without the coffee, the insomnia would probably not occur. That gives us evidence that the coffee made a difference.
Want one paramount example of contributory causation?
Climate change, with all its contributory causes: sun, orbital variations of the earth, low cloud cover, oceans, albedo effect, man-made C02, etc.
REMEMBER, VERY IMPORTANT...
Deductive and Inductive arguments
PREMISE:
A premise is a reason given.
ARGUMENT:
An argument is a set of premises and a conclusion.
An argument is a set of premises and a conclusion. Look at this argument:
"Socrates is a man" first premise
"Men are mortal" second premise
________
"Socrates is mortal" conclusion
Now, there are deductive and inductive arguments.
Deductive argument are “truth preserving”, because the truth of its premises guarantees the truth of its conclusion.
Deductive arguments can be valid or invalid. If VALID, then the conclusion necessarily follows from the premises (even if the premises were false).
"Socrates is a man" (T)
"Men are mortal" (T)
________
"Socrates is mortal" (T)
the above argument is valid. in addition the premises are true. if the argument is VALID and its premises true, the conclusion must be true. we call this kind of argument SOUND.
now let's play a bit with truth values:
"Socrates is a man" (T)
"Men are immortal" (F)
__________
"Socrates is immortal" (F)
the above argument IS VALID, i.e, though the conclusion follows from the premises, though one premise is false. We have a false premise and a false conclusion, yet the above argument is VALID, but UNSOUND
"Socrates is a man" (T)
"Men are immortal" (F)
__________
"Socrates is immortal" (F)
the above argument IS VALID, i.e, though the conclusion follows from the premises, though one premise is false. We have a false premise and a false conclusion, yet the above argument is VALID, but UNSOUND
See that deduction is independent from experience beause the reasoning is self contained in the premises. Remember that the premises could be false and the argument still be valid.
DEDUCTION IS APRIORISTIC. MATHEMATICS IS AN EXAMPLE OF AN APRIORISTIC DISCIPLINE.
______________
INDUCTIVE ARGUMENTS:
An inductive argument establishes a conclusion with a high degree of probability. Here the truth of their premises does not guarantee the truth of their conclusion.
Every windstorm observed so far in this area comes from the north. We can see a big cloud of dust in the distance. So, a new windstorm is probably coming from the north.
Analysis: The above argument is a strong inductive argument. The "so far" in the premise is simple and cautious, and the conclusion uses "probably."
Strong inductive arguments establish the conclusion with high probability IF the premises were
true.
If the argument is not STRONG, then it is WEAK.
THRE ARE THREE KINDS OF INDUCTIVE ARGUMENTS:
Inductive generalization: It goes from a premise about a sample to a conclusion about the population, or it derives general principles from specific observations. Click here for a good explanation of inductive generalizations.
Analogical Induction: The process of analogical inference involves noting the shared properties of two or more things and from this basis, inferring that they also share some further property:
Example:
Athenians and spartans are similar with respect to speaking the same language & being Pantheists
An Athenian has been observed to have further property X
Therefore, a Spartan probably also has property X.
An Athenian has been observed to have further property X
Therefore, a Spartan probably also has property X.
Enumerative Induction: It reasons from particular instances to all instances.
Example: If one observes 100 swans, and all 100 are white, one might infer "All swans are white."
(As you can see, even if the premises are true, it does not entail the conclusion's truth. The conclusion might be true, and probably true, yet it can be false).
Inference to the best explanation IBE:
Phenomenon Q.
E provides the best explanation for Q.
Therefore, it is probable that E is true.
________________
As you can see, a big difference between deduction and induction is that the latter depends from experience.
INDUCTION IS BASICALLY A POSTERIORI.
SO, WE CAN SAY THAT SCIENTIFIC THEORIES ARE A POSTERIORI, NEVER ABSOLUTELY CERTAIN.
Necessary and sufficient conditions
Necessary conditions:
X is a necessary condition for Y,
if X is not present, Y doesn't happen.
(yet, to say that X is a necessary condition for Y does not mean that X guarantees Y)
Having gasoline in my car (gasoline engine cars) is a necessary condition for my car to start. Yet, having gasoline in the car does not guarantee that my car will start. There are many other conditions needed for my car to start.
Having oxygen in the earth's atmosphere is a necessary condition for human life. However, having oxygen will not guarantee human life. There are many other conditions needed for human life other than oxygen in the atmosphere.
Sufficient conditions:
X is a sufficient condition for Y,
if X is present, Y happens (X guarantees Y)
Rain pouring from the sky is a sufficient condition for the ground to be wet (not necessary, since the ground could be wet for other reasons).
______
Test yourself:
*Is sunlight a necessary or sufficient condition for the flowers to bloom?
*Is earning a final grade of C a necessary or sufficient condition for passing the course?
*Is being a male a necessary or sufficient condition for being a father?
*Is having AIDS a necessary or sufficient condition for having the HIV virus?
*Is studying for a test a necessary or sufficient condition for passing a test?
*Is completing all the requirements of your degree program a necessary or sufficient condition for earning your degree?
Thursday, August 27, 2026
la naturaleza de la HIPÓ-TESIS
Henri Poincaré
La hipótesis (hypo: debajo, thesis: conclusión) es el recurso par excellence del descubrimiento científico.
"Siendo incierta la causa de las cosas, no es indigno del hombre conjeturarla; mas tenga por cierto solamente aquello que los hechos consienten." Anónimo, siglo I AD
"Así como el peregrino, viendo humo en lontananza, presume que allí ha de haber fuego, así el entendimiento, viendo las señales del mundo, propone lo que ellas pudiesen significar. Mas sólo Dios conoce la verdad." Fray Fabrenius, circa 1300 AD
"No debe llamarse verdad aquello que apenas ha sido supuesto para explicar los fenómenos; pero tampoco ha de despreciarse una hipótesis que, sometida a examen, ordena lo diverso y hace inteligible lo que antes parecía confuso." George Hamman, siglo 17.
Recuérdese. Hypótesis no es conclusión definitiva. Es meramente postulado plausible.
La hipótesis necesita confirmación. Esa confirmación se da en las matemáticas en forma absoluta (no así en las demás ciencias experimentales).
¿Qué garantías tenemos de que nuestra hipótesis funcione?
De esto vamos a hablar muy pronto.
Some excellent introductory math texts I recommend
Mi libro favorito de cálculo 1, 2 y 3. Tom Apostol Vol. 1, & Volume 2 (PDF).
Lynn Loomis and Schlomo Sternberg, for Advanced Calculus. What a mind, Svi Sternberg!
A great Linear Algebra Introduction (PDF).
Introduction to Abstract Algebra, by Michael Artin (2nd Edition).
Differential Equations, PDF, by Paul Blanchard. Professor Blanchard is an authority on the Mandelbrot set.
Applied Partial Differential Equations, David Logan, PDF.
Introduction to Set Theory, by Herbert Enderton, PDF.
Monday, August 24, 2026
LES PRESENTO EL AXIS DEL CONOCIMIENTO
In the Y axis, I know that I know (certainty)
In the X axis, I know I don't know (doubt, duda, la filosofía)
In the -Y axis, I don't know, I know (self deceit, autoengaño)
In the -X axis, I don't know I don't know (blissful ignorance or self-deceit)
3. "No sé que sé"
Aquí las preguntas suelen revelar un saber que ya estaba operando
¿Cómo sabes que esta persona está mintiendo? «No sé… simplemente me doy cuenta.» Hay un saber, pero no está tematizado. El cocinero: ¿Cómo sabes hacer esto? «No sé cómo; simplemente puedo.» Se le llama saber práctico o tácito. ¿Cómo supiste que algo no estaba bien cuando entraste en la habitación? «No sé; lo sentí inmediatamente.» Reconocimiento sin una formulación consciente de las razones.
Y una pregunta más sartreana (a una joven cajera del Presidente): ¿Sabes que podrías dejar este trabajo y matricular el college? «No, tengo que seguir aquí.» La pregunta apunta a un saber sobre la propia libertad que el sujeto no quiere reconocer como saber.
Distingamos dos formas de 3:
3a. No sé que sé → saber tácito.
3b. No sé que sé → saber rechazado/negado.
4. "No sé que no sé"
Aquí las preguntas son mucho más extrañas, porque la persona no sabe todavía cuál es la pregunta que debería hacer. Ejemplos:
1- Un niño pregunta: «¿Por qué el cielo es azul?» Eso todavía es 2: sé que no sé. Pero antes de descubrir que existe una explicación física del color del cielo, podría ni siquiera preguntarse por qué tiene color. Ese sería el 4.
2- Un hombre del siglo XV: «¿Qué causa las enfermedades?» Puede formular la pregunta. Pero no puede preguntar todavía: «¿Qué microorganismo está causando esta infección?» No dispone del concepto de microorganismo.
3- Antes de Copérnico, alguien puede preguntar: «¿Por qué se mueven los planetas?» Pero difícilmente: «¿Qué evidencia demostraría que la Tierra gira alrededor del Sol?» Esa posibilidad no forma todavía parte de su horizonte conceptual.
4- Una persona que nunca ha oído hablar de inteligencia artificial no pregunta: «¿Qué efectos tendrá la IA sobre mi profesión?» No sabe todavía que existe esa cuestión.
Pregúntale a alguien: «¿Qué no sabes?»
Si responde: «No sé cálculo diferencial.» Está en 2.
Pero si le preguntas: «¿Qué cosas ignoras cuya existencia ni siquiera sospechas?»
Ya estás señalando 4.
Y ahí está la diferencia profunda:
El 2 puede formular su ignorancia.
El 4 no puede formularla porque todavía carece del objeto de su ignorancia.
Por eso 4 es el más cercano a la ignorancia absoluta, mientras que el 2 es ya una forma de conocimiento.