Logic: empirical or abstract science?

 
Speakpigeon
 
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Speakpigeon
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14 January 2019 10:25
 

There is an important distinction to be made between two conceptions of logic as an empirical or an abstract science.

Mathematicians who worked on the conception of a method of logic in the 19th century, Frege in particular, were essentially and explicitly motivated by the idea that a proper method of formal logic would help improve the rigour of mathematical proofs, a particular concern at the time, between the two extremes of Abel and Weierstrass. This suggests a view of logic as essentially not arbitrary and therefore as essentially empirical.

And in effect, mathematicians working on a method of logic at the time had to rely on the only empirical evidence available to them, i.e. Aristotle’s syllogistic theory, plus what other people since had said on the subject, including other mathematicians, as well as their own personal intuition, as to what formulas could be accepted as logical truths, this in order to work out a method of logical calculus they could use to improve rigour of proof.

Today, on the surface, we seem to have a very different perspective, whereby logic is more often understood as essentially a mathematical object, like the set of Real numbers is, so that logic is thought of as being the methods of logic themselves that mathematicians have contrived since Frege. In this perspective, logic is no longer seen as an essentially empirical science, but as the motley collection of theories, seen as arbitrary at least in principle, that mathematicians are working on as objects of study rather than as methods they could use to improve the rigour of proofs.

Meanwhile, mathematicians themselves still essentially use and effectively rely on their own, intuitive, sense of logic to prove theorems, producing what can be described in effect as semi-formal proofs.

The few examples of formal logic being used to prove theorems today all rely on some variation of Gentzen’s “natural” method of proof (conceived between 1929 and 1935), which is essentially a modern generalisation of Aristotle, and a method which effectively relies on the crucial use of so-called rules of inference, which are formulas all essentially taken from the set of formulas long recognised as logical truths in the Aristotelian tradition, save a few exceptions.

So, in effect, all current practice of mathematical proof, be it intuitive or making use of theorem provers, like Isabel in Germany and Coq in France, still literally relies ultimately on the empirical evidence available to mathematicians that some logical truths are evidently true. Yet, the fundamentally empirical nature of the logic practised by mathematicians themselves, today as always since Euclid, is somewhat airbrushed out of the picture in favour of a more abstract notion of it.

Comments welcome.
EB

[ Edited: 14 January 2019 10:36 by Speakpigeon]
 
Abel Dean
 
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Abel Dean
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17 January 2019 21:13
 

You seem to be using the word, “empirical,” as having a meaning at odds with the way I have always defined “empirical.” It is empirical if it less a matter of abstraction and more a matter of observations of the physical reality external to our minds. Mathematical proofs of any sort are not empirical, as I understand that word.

 
Speakpigeon
 
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Speakpigeon
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18 January 2019 04:46
 
Abel Dean - 17 January 2019 09:13 PM

You seem to be using the word, “empirical,” as having a meaning at odds with the way I have always defined “empirical.”


I wouldn’t know, I don’t know you that well.
You differ on how you use some word. And?
Not to worry, it happens a lot.
Still, it’s of course not for you to provide a common definition of the meaning of words as they are used by native speakers.
So, I guess, I don’t see where would be the problem. Change your way of life?

Abel Dean - 17 January 2019 09:13 PM

It is empirical if it less a matter of abstraction

Abstraction? What’s that? Do you happen to have any empirical evidence of what you call “abstractions”?
I am aware that empirical scientists would talk of all the concrete trees and all the concrete flowers they have evidence for as somehow belonging to something really very abstract called “the genera of plants”, where I myself can only see the one tree in my garden and a few flowers in the garden across the street.
The thing is, I’m not too sure where these people get their evidence of any “genera of plants”. You tell me.

Abel Dean - 17 January 2019 09:13 PM

and more a matter of observations of the physical reality

Ah. Well. There you go. I should be berating you for making stuff up. Not that you’re the only one around here, but. So, no. You need to be told that what the definition of the meaning of words as they are used by native speakers is a matter of empirical evidence. Yes, I know, even for the word “empirical” itself. So, first, you need to know where to look for evidence in this particular instance. You clearly believe it’s good enough to trust yourself as the expert on such things but you shouldn’t. You’re not expert just for being a native speaker. You need to look at a dictionary. So, here it is:

empirical
adj.
1.
a. Relying on or derived from observation or experiment: empirical results that supported the hypothesis.
b. Verifiable or provable by means of observation or experiment: empirical laws.
2. Guided by practical experience and not theory, especially in medicine.

That’s the empirical evidence for what the word “empirical” really means.
I guess you can see for yourself that it doesn’t say here that empirical is a matter of “observations of the physical reality”, as you unfortunately asserted.
So, me I use definition 1a here. You? Ah, yes, you use your own made-up definition. So no surprise we should differ. Problem solved.

Abel Dean - 17 January 2019 09:13 PM

external to our minds.


Minds? What’s that again? Do you think we have empirical evidence of “our minds”?
Or do you mean that the mind is a physical reality that we don’t have empirical evidence of?
Only you know.

Abel Dean - 17 January 2019 09:13 PM

Mathematical proofs of any sort are not empirical, as I understand that word.

Well, I hope you’ve learned something, then.
Like, to each kind of reality, its empirical evidence, if any.
If you did go to Heaven one day, what would you do? You would… not believe it?
Oh, wait, I know, you don’t have empirical evidence of the existence of logic.
EB

[ Edited: 18 January 2019 04:51 by Speakpigeon]
 
Abel Dean
 
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Abel Dean
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18 January 2019 19:22
 

I have a rough idea, but I would like to more completely understand what you are saying. Maybe it would help if you would give two simple examples: an empirical proof and a non-empirical proof. I am sorry if you already covered this in your previous lectures. I am new to the forum.

 
Speakpigeon
 
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19 January 2019 02:07
 
Abel Dean - 18 January 2019 07:22 PM

I have a rough idea, but I would like to more completely understand what you are saying. Maybe it would help if you would give two simple examples: an empirical proof and a non-empirical proof. I am sorry if you already covered this in your previous lectures. I am new to the forum.

Yes, I can tell you’re new just because you’re definitely polite and you don’t pretend you know all there is to know. Most people here start by taking you like shit, posting irrelevant comments, and throwing the occasional tantrum. So, thanks.
Rather than replying here, I’ll start a new thread on empirical evidence, since it seems an interesting topic in itself.
EB