Space Cadet Will Hutton is sadly misinformed.

[AndySir]

I believe everyone understands the experience of intuition. However anyone with any intelligence whatsoever must recognise that the truth is often counter-intuitive and that that voice is unreliable. Observation is good, it is simply collecting data. Even psychologists have abandoned introspection.

I was under the impression I was making my own point, independent of discussions of moon landing hoaxes, so I'm not sure how I could have missed it.

[JavaScriptDonkey]

I believe everyone understands the experience of intuition. However anyone with any intelligence whatsoever must recognise that the truth is often counter-intuitive and that that voice is unreliable. Observation is good, it is simply collecting data. Even psychologists have abandoned introspection.

I was under the impression I was making my own point, independent of discussions of moon landing hoaxes, so I'm not sure how I could have missed it.

An appeal to common sense is often the last gasp of a unsupported argument drowning in a sea of facts that the proposer simply doesn't understand. Common sense is nothing more than something people without actual knowledge console themselves with. It is subjective in the extreme.

Intuition & introspection as tools of scientific discovery went out with Aristotle. They are still useful when considering actual data but by themselves carry no more weight than a mild hunch.

[caspian]

Anyone who can provide an example of some piece of knowledge or awareness about the universe which has been derived from a sense of deep human motivation, common sense or even pure reason I invite to do so.

What about pure mathematics? Mathematicians can arrive at truth through pure reasoning. Mathematics is probably the only subject by which absolute truths can be found - other sciences can only ever arrive at an approximation of how the universe works.

[Ludwig]

An appeal to common sense is often the last gasp of a unsupported argument drowning in a sea of facts that the proposer simply doesn't understand. Common sense is nothing more than something people without actual knowledge console themselves with. It is subjective in the extreme.

I have plenty of actual knowledge, and I combine it with intuition and introspection to try to understand the world.

Common sense and intuition can be honed - you test what actually happens against what your gut feeling was, and if there's a discrepancy, you adjust your mental model of the world.

I'm not claiming I know what intuition is (actually I think it's a mixture of things), but I am satisfied in my own mind that it exists, and that it is a reliable way of understanding things in essence, as opposed to in detail.

Still, I'm an intuitive person, and these things come naturally to me.

Intuition & introspection as tools of scientific discovery went out with Aristotle. They are still useful when considering actual data but by themselves carry no more weight than a mild hunch.

Etymologically, "science" just means "knowledge".

I have learned just as much through introspection and intuition - and literature - as through reading about science. (And I read about science probably as much as most people on this forum.)

Understanding the universe isn't just about facts, it's also about ideas and patterns. These patterns - for example to people's lives and personalities, to history - are perceptible by intuition and observation, but are tyically too complex to be explained by empirical science. It is possible to grasp general patterns and tendencies by intuition - whatever intuition is, exactly.

One intuition I've always had is that our civilisation is entering its last phase - obtained just from a general sense of ennui in the world. While scientists were all promising a wonderful future of space travel and living till 300, I was thinking it all sounded like bollox, and I take a bitter satisfaction in seeing that I was right.

[AndySir]

Anyone who can provide an example of some piece of knowledge or awareness about the universe which has been derived from a sense of deep human motivation, common sense or even pure reason I invite to do so.

What about pure mathematics? Mathematicians can arrive at truth through pure reasoning. Mathematics is probably the only subject by which absolute truths can be found - other sciences can only ever arrive at an approximation of how the universe works.

No. All mathematics is based on physics and geometry. The founding axioms (calculus, for example) are scientific models. Research mathematicians follow a pretty standard hypothesis, proof (experimentation), refutation model, although I take your point that the a mathematical proof is far more demanding than one of the sciences.

Mathematicians may go off building castles in the sky, but they do take care to check and refine their assumptions in the real world so that they can fill in the foundations later. Without that mathematics is a game, the study of which could be compared to the study of chess. Entertainment.

[caspian]

No. All mathematics is based on physics and geometry. The founding axioms (calculus, for example) are scientific models.

Well, I'm not a mathematician, but pure mathematics isn't based on physics. The axioms are assumptions about mathematical principles that cannot be proved by deduction, e.g. (a + b) = (b + a).

[caspian]

One intuition I've always had is that our civilisation is entering its last phase - obtained just from a general sense of ennui in the world.

The trouble with that is that every single generation has had its share of people who thought that civilisation was coming to an end. I dare say that those who lived through the First and Second World Wars might've had good reason to think that it was the end of everything. How do you know that your intuition is any more valid than all those people before you who thought the same thing?

One day, someone might be right, but that's just the law of averages.

[AndySir]

No. All mathematics is based on physics and geometry. The founding axioms (calculus, for example) are scientific models.

Well, I'm not a mathematician, but pure mathematics isn't based on physics. The axioms are assumptions about mathematical principles that cannot be proved by deduction, e.g. (a + b) = (b + a).

Bad example. Commutativity can be shown to a four year old. Take a of any element (apples) and and b to it. Count elements. Mathematical axiom clearly based on observation. Clearly not deduction but I'm assuming that was an error as it would just leave us with a miserable tautology for an argument.

[DominicJ]

Physics is too complicated for phyisists.
Or so my Pure Maths Textbook said

[caspian]

Commutativity can be shown to a four year old. Take a of any element (apples) and and b to it. Count elements. Mathematical axiom clearly based on observation.

That's just an example of the axiom, not a mathematical proof, which isn't possible (hence why it's an axiom). A rigorous mathematical proof is immune to any future disproof, unlike scientific theorems, which can always (in theory at least) be disproved.

[AndySir]

Commutativity can be shown to a four year old. Take a of any element (apples) and and b to it. Count elements. Mathematical axiom clearly based on observation.

That's just an example of the axiom, not a mathematical proof, which isn't possible (hence why it's an axiom). A rigorous mathematical proof is immune to any future disproof, unlike scientific theorems, which can always (in theory at least) be disproved.

So your argument for pure mathematics being based on reason and not being based on empiricism is that its founding assumptions can only be based on empiricism and not reason?

It appears you did mean the tautology I assumed was a mistake before. If you could prove your initial assumptions then they wouldn't be assumptions, so saying you can't prove them is hopelessly circular.

[Ludwig]

One intuition I've always had is that our civilisation is entering its last phase - obtained just from a general sense of ennui in the world.

The trouble with that is that every single generation has had its share of people who thought that civilisation was coming to an end. I dare say that those who lived through the First and Second World Wars might've had good reason to think that it was the end of everything. How do you know that your intuition is any more valid than all those people before you who thought the same thing?

Oh, the number of times I've had to answer this question!

The most obvious answer is that, this time, the facts - the scientific facts that you trust - all point to a coming cataclysm.

The scientist looks at the facts and says, "There's no reason why we can't adapt - with some discomfort, sure, but not too much pain." But this analysis discounts human nature, discounts the fact that in times of hardship, people's concern for themselves and their families overrides concern for humanity at large.

These are not things that can be inferred from the quantitive scientific method, but they are self-evident to anyone who's studied human behaviour and knows a bit of history.

When resources are scarce, people always end up fighting each other for them. This is the message of "Collapse". Not everyone fights, of course - but those who don't fight are the first to lose out.

But to return to the question of intuition: a study of history and biography, as well as literature, illustrates the ways in which people and empires that are tired and that have peaked culturally go into decline. In the cases of people, they often get literally ill; but even if not, the vigour goes out of them, they lose their motivation. Societies in a similar state descend first into hedonism, then into incoherence, and finally into collapse.

It's a pattern, but not a course of events that can be predicted by scientific analysis.

If you were to deny that the past is any kind of guide to the future, I couldn't prove you wrong, but I'd think you were a fool.

One day, someone might be right, but that's just the law of averages.

My instincts have proved right on many occasions. While I agree that many people's, perhaps most people's, are usually wrong or misguided or misinterpreted, experience leads me to take my own seriously, especially when they are backed up by empirical evidence.

But I can see it's pointless to try to convince you about this.

[caspian]

So your argument for pure mathematics being based on reason and not being based on empiricism is that its founding assumptions can only be based on empiricism and not reason?

I'm afraid you've lost me there. My original point wasn't about the axioms at all, it was that pure mathematics doesn't rest on physical laws.

[biffvernon]

The trouble with that is that every single generation has had its share of people who thought that civilisation was coming to an end.

Here is the relevant section taken from the late David Fleming's book, Lean Logic, which will be published on the 7th of July:

Wolf, The. The fallacy that, since previous warnings of a problem have been wrong, or premature, or misunderstood, they must be wrong now.

One of Aesop’s Fables is the story of the boy whose job was to look after the sheep but, having a nervous disposition, he was forever crying “wolf” when no wolf was there. One day the wolf really did come, and he cried “wolf” again, but nobody believed him, and the wolf was able to dine off the sheep and the boy at leisure.

There are two morals to the story. The first is: avoid giving false alarms. The second is: in the end, the wolf came, so do not be misled by previous false alarms into thinking that the latest alarm is false, too. Of these two morals, the second one is more significant. Believing false alarms wastes time, but it can lead to some helpful advice for apprentice shepherds; disbelieving all alarms can lead to a local lad being eaten, for starters.

We have an example of the fallacy of the wolf in the case of supplies of oil. A century or so ago, there were some false alarms about how little oil remained; the art of forecasting oil supplies earned a bad reputation. However, estimates of the quantity remaining in the world, and of the turning-point (the “peak”) at which oil production would start to decline, steadily improved and, in the 1970s, estimates of the accessible and liquid oil which had been in place at the start of the industrial era settled at, or around, 2000 billion barrels, and that estimate has held. The expected peak was estimated to be around the year 2000 – later extended a few years into the new century thanks to the slower growth in demand following the oil shocks of 1973-1979. The “2000:2000” warning, starting with a report by Esso in 1970, was independently confirmed and published by official sources, such as the UK’s Department of Energy (1976), the Global 2000 Report to the President (1980), the World Bank (1981), and by numerous independent studies such as Hubbert (1977), Petroconsultants (1995), Ivanhoe (1997), Campbell (1999), Bentley (2002), and so on through the first decade of the new century. Analysts have also pointed to the regrettable consequences of a breakdown in oil supplies on a global market which has neglected to make any serious preparation. Here was a wolf that gave more than forty years notice of its arrival, and has been thoughtfully issuing reminders ever since. (For detail on recent studies, see *Energy Prospects).

It is, however, the sceptics that tend to carry the day. “There is always a series of geologists who are concerned about imminent depletion of world supplies”, an energy economist reassured a House of Lords Select Committee on Energy Supply. “They have been wrong for 100 years and I would be confident they will be wrong in the future”. So that’s all right then: the anguished warnings are nothing more than that new kid trying to draw attention to himself. Aesop might be tempted to revise his fable slightly. Here we have the apprentice shepherd growing mature and experienced in the job. He has been giving precise fixes of the wolf’s approach for as long as anyone can remember. He is specific and credible about the action that must be taken to save the village. And still he is disbelieved.

[AndySir]

So your argument for pure mathematics being based on reason and not being based on empiricism is that its founding assumptions can only be based on empiricism and not reason?

I'm afraid you've lost me there. My original point wasn't about the axioms at all, it was that pure mathematics doesn't rest on physical laws.

Okay... breaking it down as a logical argument.

1. All mathematics relies on foundation axioms.
2. All mathematical axioms are based on empiricism (observation)
3. Therefore mathematics is based on empiricism (is not purely reasoned)