Generally, there are no errors, only facts, in Euclid, but the presentation though thoroughly syllogistic, is not scientific.

Mathematics is not science. It describes logic, or philosophy. Someone must take it, and apply it to the real world to let it make sense. Proving that certain mathematics applies to certain problems in the world is part of science. The mathematics itself is not as it does not tell us how the world is or behaves.

]]>Aristotle’s study of animals is full of astute detailed observations and speculations on their relationships.

Observations and speculations are not science. Testing the speculations, or hypothesis is the final, and crucial, step. Aristotle is generally considered one of the philosophers who prepared the way for science, but his “astute observations” remain just that, observations.

However, realist classical philosophy, that is Aristotle and Aquina, is the only possibly correct account of scientific knowledge and takes natural phenomena more seriously than do the moderns.

Given that Aquinas proved the existence of God without having to resort to observables, I doubt it. Also, progress in the knowledge of the world in the 1.5 Milleniums in between the two thinkers has not been spectacular.

Besides Archimedes, the classical world, although perhaps not directly from Aristotle, offers the sphericity of the earth, Aristarchus’s heliocentric model, and the astonishing Antikythera machine.

Indeed, and these feats had nothing to do with classical philosophy, but with observations and measurements.

]]>fact and reasoned fact

For the reasoned fact, that is, scientific syllogism or proof, one has to have the appropriate object of study and the commensurate universal cause, the immediate necessary cause.

As an example, the Pythagorean theorem about right angled triangles and squares states a true fact, and is the conclusion of a valid syllogism.

A more nearly scientific fact is the construction of Pappus that, given any triangle and any parallelograms on two sides, produces a parallelogram on the third side that is the sum of the given parallelograms.

In the case of a right triangle and squares the resultant parallelogram is the usual square.

We see that right triangle and squares are not the appropriate objects, but rather any triangle and any parallelograms.

Generally, there are no errors, only facts, in Euclid, but the presentation though thoroughly syllogistic, is not scientific.

This kind of thing happens constantly in mathematics today.

]]>the reason classical philosophy was of little help in exploring the universe

Classical philosophy may have not been much or properly used in some scientific areas but it was used in others. Aristotle’s study of animals is full of astute detailed observations and speculations on their relationships.

However, realist classical philosophy, that is Aristotle and Aquina, is the only possibly correct account of scientific knowledge and takes natural phenomena more seriously than do the moderns.

It has no problem accommodating the exploration of nature from the point of view of quantity, characteristic of modern science.

Besides Archimedes, the classical world, although perhaps not directly from Aristotle, offers the sphericity of the earth, Aristarchus’s heliocentric model, and the astonishing Antikythera machine.

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Classically this is not scientific because it merely enumerates and does not mention causes.

That is the reason classical philosophy was of little help in exploring the universe. Aquinas could prove the existence of god, but not the size of the earth (which was done by explicit observation). They could argue about the essence or accidental nature of the whiteness of swans, but were utterly unable to find Australia and observe black swans.

In the end, the pinnacle of 1500 years of science in the classical world was Archimedes’ principle of water displacement. That was all of the science of the classical world that endured. What classical philosophy was good at was developing mathematics.

@modem

For example, consider mathematical induction.

Induction is good for some problems. Not every problem allows an analytical solution and a non-constructive proof is still a proof.

It will convince one of a fact, but it does not provide proofs, that is, the reasoned fact.

I do not understand what this means?

]]>If a student wants The Truth, she should switch to Mathematics

As an addendum, this isn’t true for modern mathematics, which suffers from the same neglect of causes as other sciences.

For example, consider mathematical induction. It will convince one of a fact, but it does not provide proofs, that is, the reasoned fact.

As a very instructive exercise, try establishing the formula for the sum of the first N squares by mathematical induction, and proving it by a completely causal mathematical argument.

One learns much more via the second treatment.

]]>or any treatise about the philosophy of Science

All these statements reflect the starting point of modern (Enlightenment) thinking. The starting point is merely an arbitrary.choice and is never in spite of much writing justified by its proponents.

They are all false from the classical starting point of knowledge of external things, natures, causes, essence or form, and existence. The classical starting points and development are justified in Aristotle and Aquinas.

As a trivial illustration, consider Popper’s example of falsification or scientific statement.

All swans are white.

This is falsifiable because a black swan is logically possible.

Classically this is not scientific because it merely enumerates and does not mention causes. The scientific question is whether whiteness is a part of the essence of swan. If so then all swans are white. As it clearly is not part of the essence, the statement os about an accidental situation.

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