Is the word "Empirical" meaningless?

[PseuTonym]

The word “theoretical” is helpful if we are able to identify a particular hypothesis that is being relied upon. It is a kind of disclaimer that says, “The following would be true if your hypothesis were true”, so that what comes next isn’t mistakenly taken to be a claim that “the following is true.”

[The intuitionists Brouwer and Heyting appealed to the “impossibility” of ever knowing whether and where the sequence 0123456789 appears in the decimal expansion of pi …

Yet it was found in 1997 by Kanada, beginning at position 17,387,594,880.](http://forums.catholic-questions.org/showthread.php?t=1021291)

According to the Brouwer and Heyting theory of the decimal expansion of pi, what are the digits of pi? Clearly, we aren’t supposed to see what happens beginning at position 17,387,594,880. It seems that empiricism has simply overturned the theory of Brouwer and Heyting, so that there is no Brouwer-Heyting pi that can be studied. At least, it was empiricism in the sense that data was generated, and people looked for the digit sequence 0123456789 in the data, without having a theoretical reason for believing either that the digit sequence 0123456789 would be found or that it wouldn’t be found.

On the other hand, what is the meaning of the word “empirical”?

“based on, concerned with, or verifiable by observation or experience rather than theory or pure logic.”

However, is it possible to observe what begins at position 17,387,594,880 in the decimal expansion of pi without relying upon theory?

 

[Gorgias]

what is the meaning of the word “empirical”?

“based on, concerned with, or verifiable by observation or experience rather than theory or pure logic.”

However, is it possible to observe what begins at position 17,387,594,880 in the decimal expansion of pi without relying upon theory?

You seem to be confusing ‘theory’ with ‘definition’.

One may certainly observe any of the digits of pi – given sufficient rigor and patience – simply by applying the definition of pi. There is no theory involved in “the ratio of a circle’s circumference to its diameter.”

 

[PseuTonym]

One may certainly observe any of the digits of pi – given sufficient rigor and patience – simply by applying the definition of pi.

How? For example, how could you possibly determine the 100th digit without relying upon any theory?

You can construct a physical model and perform measurements, but you won’t know whether a given measurement is greater than or less than the true value, and your measurements aren’t likely to agree with each other on more than a few digits.

 

[seagal]

How? For example, how could you possibly determine the 100th digit without relying upon any theory?

You don’t have to rely on theory, you just have to do the math. That’s not theoretical, that’s practical.

 

[PseuTonym]

You don’t have to rely on theory, you just have to do the math.

To do the math, you rely upon theory.

Are you familiar with any of the history of this topic? Heyting and Brouwer were above average as mathematicians. Their error was not a result of ignoring data that was available to them. Their error resulted from the following combination: the state-of-the-art of mathematics and computing technology of their era, plus their unusual point of view regarding the philosophy of mathematics.

I am frankly surprised by the responses to this thread. If we don’t distinguish between observation and calculation, then we aren’t in a position to recognize a discrepancy between the results of observations and the results of calculations.

 

[Charlemagne_III]

On the other hand, what is the meaning of the word “empirical”?

“based on, concerned with, or verifiable by observation or experience rather than theory or pure logic.”

Nothing wrong with this definition.

The tendency among some is to believe that empiricism trumps pure logic and somehow is superior to it.

But the pure logic of mathematics can tell us things we have not empirically observed.

The Big Bang, for example, was deduced mathematically by George Lemaitre before the empirical evidence began to appear decades later.

 

[STT]

On the other hand, what is the meaning of the word “empirical”?

“based on, concerned with, or verifiable by observation or experience rather than theory or pure logic.”

Empirical has to do with both observation and theory.

However, is it possible to observe what begins at position 17,387,594,880 in the decimal expansion of pi without relying upon theory?

You are correct with your observation.

 

[Monte_RCMS]

What is the difference between empirical and experiential?

 

[on_the_hill]

but you won’t know whether a given measurement is greater than or less than the true value, and your measurements aren’t likely to agree with each other on more than a few digits.

You seem to be leaning toward a post-positivist approach: “We can’t know the true value.” And you’re leaving off the second part of it: “But we can get close enough.”

The post-positivists don’t reject empirical analysis.

 

[PseuTonym]

Empirical has to do with both observation and theory.

Insofar as usage of the word “empirical” is concerned, you might be absolutely right. However, the official definition says “rather than theory.” If there is a conflict between usage and the official definition, then the word “empirical” is a source of confusion rather than clarity.

We see something similar with the word “islam.” Instead of a reasonable discussion, public discourse seems to proceed something like the following …

Side A: “Quran, Islam, Quran, Islam. Let there be no compulsion in religion.”

Side B: “Sharia, Islam, Sharia, Islam. The law mandates death to apostates.”

Maybe an appropriate response to philosophical claims that rely upon the conflicting meanings assigned to the word “empirical” is to ask the question: “Do you mean pure Quran empirical, pure Sharia empirical, or some combination of Quran and Sharia?”

 

[PseuTonym]

Nothing wrong with this definition.

The definition: “based on, concerned with, or verifiable by observation or experience rather than theory or pure logic.”

How about the word “verifiable”?

The root seems to be something like “verify” from the Latin “veritas” (truth). The Russian language word for truth: pravda.

Comrade Charlemagne III, verification and Pravda-ification are the same thing!

Can you provide a list of what is known to you as having received the official stamp of Pravda-ification? It would make life easier for the poor, tormented souls who seek truth.

 

[PseuTonym]

The post-positivists don’t reject empirical analysis.

Is that a reference to Sharia empirical analysis, Quran empirical analysis, or some combination of the two?

 

[inocente]

The word “theoretical” is helpful if we are able to identify a particular hypothesis that is being relied upon. It is a kind of disclaimer that says, “The following would be true if your hypothesis were true”, so that what comes next isn’t mistakenly taken to be a claim that “the following is true.”

[The intuitionists Brouwer and Heyting appealed to the “impossibility” of ever knowing whether and where the sequence 0123456789 appears in the decimal expansion of pi …

Yet it was found in 1997 by Kanada, beginning at position 17,387,594,880.](http://forums.catholic-questions.org/showthread.php?t=1021291)

According to the Brouwer and Heyting theory of the decimal expansion of pi, what are the digits of pi? Clearly, we aren’t supposed to see what happens beginning at position 17,387,594,880. It seems that empiricism has simply overturned the theory of Brouwer and Heyting, so that there is no Brouwer-Heyting pi that can be studied. At least, it was empiricism in the sense that data was generated, and people looked for the digit sequence 0123456789 in the data, without having a theoretical reason for believing either that the digit sequence 0123456789 would be found or that it wouldn’t be found.

On the other hand, what is the meaning of the word “empirical”?

“based on, concerned with, or verifiable by observation or experience rather than theory or pure logic.”

However, is it possible to observe what begins at position 17,387,594,880 in the decimal expansion of pi without relying upon theory?

Imho the problem is that the word ‘theory’ has more than one meaning, and you’re mixing them up. Going by the OED (link below), you’re using definition 1.2 here, but ought to be using definition 1 in science and 1.3 in math.

en.oxforddictionaries.com/definition/theory

 

[inocente]

The tendency among some is to believe that empiricism trumps pure logic and somehow is superior to it.

But the pure logic of mathematics can tell us things we have not empirically observed.

The Big Bang, for example, was deduced mathematically by George Lemaitre before the empirical evidence began to appear decades later.

Lemaitre would tell you it’s the exact opposite. Theoretical physicists write valid math all day every day, but to get anywhere they have to do the much harder task of finding the one equation which matches what nature does. As the theoretical physicist Richard Feynman said, “It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong”.

 

[on_the_hill]

Is that a reference to Sharia empirical analysis, Quran empirical analysis, or some combination of the two?

It’s a reference to the scientific method of making observations, forming a theory, forming a hypothesis, gathering data, analyzing data, and drawing conclusions.

Maybe you and I aren’t on the same page. :ehh:

 

[PseuTonym]

You seem to be leaning toward a post-positivist approach: “We can’t know the true value.”

I was attempting to distinguish an estimate based on
#1 direct measurement of a physical diagram,
… from an estimate based on …
#2 theory.

Theory made it possible for Archimedes to know that the estimate 22/7 is greater than the true value of pi.

If the best estimate of pi were based on direct measurement of a physical diagram, then how would it be possible to know that the estimate is greater than the true value of pi?

 

[on_the_hill]

I was attempting to distinguish an estimate based on
#1 direct measurement of a physical diagram,
… from an estimate based on …
#2 theory.

Theory made it possible for Archimedes to know that the estimate 22/7 is greater than the true value of pi.

If the best estimate of pi were based on direct measurement of a physical diagram, then how would it be possible to know that the estimate is greater than the true value of pi?

Well…your question stretches beyond my education in these philosophical subjects. Pi has no end point. Therefore, it’s true value doesn’t exist.

 

[PseuTonym]

Well…your question stretches beyond my education in these philosophical subjects.

If that is a polite goodbye explaining why you aren’t interested, then I thank you for the polite goodbye.

However, if you are interested, but concerned that you “aren’t qualified” to participate in this thread, then you should continue to participate in this thread.

Have you heard of Ramanujan? He was considered to be uneducated in mathematics, but he made major contributions to the subject. Coincidentally, some of his contributions involved formulas for computing pi.

Regarding Ramanujan’s formal education:

The university administration said their regulations didn’t allow a graduate scholarship to be given to someone like Ramanujan who hadn’t finished an undergraduate degree.

Regarding the letter that Ramanujan sent to G. H. Hardy:

So what was Hardy’s reaction? First he consulted Littlewood. Was it perhaps a practical joke? Were these formulas all already known, or perhaps completely wrong? Some they recognized, and knew were correct. But many they did not. But as Hardy later said with characteristic clever gloss, they concluded that these too “must be true because, if they were not true, no one would have the imagination to invent them.”

Bertrand Russell wrote that by the next day he “found Hardy and Littlewood in a state of wild excitement because they believe they have found a second Newton, a Hindu clerk in Madras making 20 pounds a year.” Hardy showed Ramanujan’s letter to lots of people, and started making inquiries with the government department that handled India. It took him a week to actually reply to Ramanujan, opening with a certain measured and precisely expressed excitement: “I was exceedingly interested by your letter and by the theorems which you state.”

Regarding formulas for computing pi:

Previous approximations to pi had in a sense been much more sober, though the best one before Ramanujan’s (Machin’s series from 1706) did involve the seemingly random number 239:

Link to the article about Ramanujan that includes a formula involving “the seemingly random number 239”

But Ramanujan’s series—bizarre and arbitrary as they might appear—had an important feature: they took far fewer terms to compute pi to a given accuracy. In 1977, Bill Gosper—himself a rather Ramanujan-like figure, whom I’ve had the pleasure of knowing for more than 35 years—took the last of Ramanujan’s series from the list above, and used it to compute a record number of digits of pi.

 

[Charlemagne_III]

Lemaitre (correcting Einstein’s math) set the mathematical foundations for the Big Bang theory long before it was verified by decades of empirical observation.

Einstein set the foundations for nuclear physics decades before the A-Bomb was invented to prove his theory.

The greatest inventions/discoveries in human history began with abstract theory and were followed by empirical observations, including the theory of evolution which is itself still evolving as new empirical data appear.

Likewise humans may contemplate the existence of God by rational investigation long before they are able to prove their speculations by encountering God in Person.

Faith is a wonderful thing, and the whole world needs it because without it we would know little for certain from our speculations. But philosophical speculations about human destiny and God should never be treated with contempt, since God gave us reason and curiosity about ultimate things, just as he gave us reason and curiosity about the things of this world and how to know them better.

 

[PseuTonym]

You seem to be confusing ‘theory’ with ‘definition’.

One may certainly observe any of the digits of pi – given sufficient rigor and patience – simply by applying the definition of pi.

Do you have any evidence that 100 decimal places of pi were computed before 1706?

Do you claim that Archimedes was aware of Machin’s formula and simply didn’t have as much patience as Machin?

Machin-like formulae are a popular technique for computing pi to a large number of digits. They are generalizations of John Machin’s formula from 1706 …

pi/4 = (4 times arctan(1/5)) - arctan(1/239)

… which he used to compute pi to 100 decimal places.

Link:
en.wikipedia.org/wiki/Machin-like_formula