Member Rara Avis
While I don't think we disagree on much, I do think there's some misunderstanding over "my" definitions. That's my fault for using "Hard" and "Soft," which pretty much already have meaning to those in the field. My definition of Hard, however, is much harder than most would ascribe to. Hard means we know we're right and there is no possibility of new research contradicting our conclusions. Your example using Fermat's last theorem is such a case. Hard, to me, means it is based on incontrovertible mathematics. It is 100 percent predictable, because 2 + 2 always equals 4.
My definition of "Soft," however, contains most of what you might consider Hard. For example, you list Biology as a Hard science, but most of it is based on experimentation and is beyond mathematical representation (with the obvious exceptions of genetics and chemistry). Mitosis is observable and well understood, and it's unlikely a new discovery could contradict what we already know. Unlikely, but nonetheless possible. To use JP's words from the other thread, just because something has happened several million times in the past is no guarantee it will happen tomorrow. In contrast, we know no number greater than three will solve Fermat's equation, and we know there will be no new number discovered tomorrow to prove us wrong. The predictability of Soft science can be very, very high, but we can never guarantee it to be 100 percent - as we can guarantee with Hard science.
My Gooey science classification probably coincides more closely to your Soft definition. Predictability is not high, for a large variety of reasons. Essentially, we are making educated guesses.
In the earlier thread, Trevor was willing to concede that 2 + 2 equals 4, opening the door to my contention that "some" realities are not a matter of perception. I knew I should have waited for Brad.
Still, I think it's clear that Brad's argument is semantical, rather than mathematical. You can debate the meaning of the words all day long, but 2 + 2 is still going to equal 4 when you crawl into bed tonight.
Except - maybe not.
Our mathematical system hasn't changed in a very long time, but it is nonetheless an evolutionary attempt to model reality. In the beginning, were positive integers. Next came negative numbers, then fractions. Interestingly, the concept of zero didn't come until much later. The ancient Egyptians never used a zero symbol in writing their numerals. Nor, more surprisingly, did the Greeks until about 1500 BC. The concept of nothing being a number was apparently not an easy leap to make.
The inclusion of zero into our system, however, produced a conundrum. If 2 times 0 equals zero, then what does 2 divided by 0 equal? Many still insist the answer must be infinity. The opposite of zero, after all, is infinity. And consider the series: 10/10=1 10/5=2 10/1=10 10/.5=20 10/.05=200 10/.005=2000. It seems evident that the smaller the divisor, the larger the result.
To this day, many competent scientists make the mistake of assuming that X/0 equals infinity. Most of Einstein's Special Relativity equations (from which the ultimate result was the famous e=mC^2), include the divisor quantity C^2 - V^2, where C is the speed of light in a vacuum and V is the velocity of matter. (Note the ^ symbol represents exponents, typically pronounced e=mc squared). Einstein's equations proved that as velocity increases, so too does the length and mass of matter, something that may be counter-intuitive but has been amply demonstrated in particle accelerators. But notice that when V exactly reaches the speed of light, C^2 - V^2 will then equal zero. Many laymen will tell you that when a particle reaches the speed of light its mass will become infinite, a blatant impossibility (even the mass of the Universe is finite). Ergo, one can never travel faster than the speed of light.
But division by zero does NOT equal infinity. Consider the following very simple equations.
(a).(a) - a.a = a2 - a2
for any finite a. This can be written as
a(a-a) = (a-a)(a+a)
dividing both sides by (a-a) gives
a = 2a
now, dividing by a gives
1 = 2, Voila!
In other words, the result of division by zero is anything you want it to be! Which is exactly why it is not allowed in mathematics. If we allow people to divide by zero, everything else in our mathematical system gets thrown out with the dirty water. So, very simply, it is forbidden.
A bit over 3,500 years ago, Mankind discovered zero because there existed real-world phenomenon that made it necessary. Indeed, every single addition to our system has been the result of the need to reflect reality, which is precisely why mathematics can be used to define reality - beyond our perceptions.
Interestingly, Einstein's equations may yet lead us to accepting the bitter necessity of dividing by zero. It was his equations (from the General Theory, this time) that predicted the Black Hole, a singularity with such dense mass that light cannot escape its gravity well. Mmmm. Einstein also showed us that gravity and acceleration are indistinguishable, so anything that can suck in light would also accelerate mass beyond the speed of light. It would, in effect, divide by zero. That's why Stephen Hawking says you "could" find a dragon flying out of a black hole. Because, when you divide by zero, anything is possible.
So, yea, Brad - 2 + 2 could equal 5. But only within the confines of a singularity.