The use of fixed point arithmetic can be appropriate under certain circumstances. Generally for scientific computing (at least in the sense that most people think of it) it is not appropriate due to the need for expressing the large dynamic ranges encountered. You mention eigenvalue problems as an example, but very often in science, one is interested in the smallest eigenvalues of a matrix (say, in computing the ground state of a quantum system). The accuracy of small eigenvalues will generally be quite deteriorated relative to large eigenvalues if you use fixed point. If your matrix contains entries which vary by large ratios, the small eigenvalues might be completely unexpressable in the working precision. This is a problem with the representation of numbers; these arguments hold regardless of how you do the intermediate computations. You could possibly work out a scaling to apply to the computed results, but now you've just invented floating point. It is easy to construct matrices whose elements are well behaved, but whose eigenvalues are exceedingly poorly behaved (like Wilkinson matrices, or even matrices with entirely integer entries). These examples are not as pathological as they might seem, and many problems at the cutting edge of science involve very poorly behaved matrices, so using fixed point in this context is a Bad Idea(TM).
You might argue that you know the magnitude of the results and you want to not waste bits on the exponent, so let's talk about the intermediates. Using fixed point will generally exacerbate the effects of catastrophic cancellations and roundoff unless you really go through great pains to work in higher precision. The performance penalty would be huge, and I would conjecture that using a floating point representation with the same mantissa bit width would be faster and more accurate.
One area where fixed point can shine is in certain areas of geometric computing. Especially if you need exact arithmetic or know the dynamic range of all the numbers beforehand, fixed point lets you take advantage of all of the bits in your representation. For example, suppose you wanted to compute the intersection of two lines, and somehow the endpoints of the two lines are normalized to sit in the unit square. In this case, the intersection point can be represented with more bits of precision than using an equivalent floating point number (which will waste bits on the exponent). Now, it is almost certainly the case that the intermediate numbers required in this calculation need to be computed to higher precision, or at least done very carefully (like when dividing the product of two numbers by another number, you need to be very careful about it). In this respect, fixed point is advantageous more from the representation standpoint rather than from a computational standpoint, and I would go so far as to say this is generally true when you can establish definite upper and lower bounds on the dynamic range of your algorithm outputs. This happens rarely.
I used to think that floating point representations were crude or inaccurate (why waste bits on an exponent?!). But over time I've come to realize that it really is one of the best possible representations for real numbers. Things in nature show up on log scales, so real data ends up spanning a large range of exponents. Also to achieve the highest possible relative accuracy requires working on log scales, making the tracking of an exponent more natural. The only other contender for a "natural" representation is symmetric level index. However, addition and subtraction are much slower in that representation, and it lacks the hardware support of IEEE 754. A tremendous amount of thought was put into the floating point standards, by a pillar of numerical linear algebra. I would think he knows what the "right" representation of numbers is.