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Roger Penrose writes in his book The Road to Reality [1] that,

Einstein suggested the following:

One can give good reasons why reality cannot be represented as a continuous field...
Quantum phenomenon...must lead to an attempt to find a purely algebraic theory 
for the description of reality. But nobody knows how to obtain the basis of
such a theory.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Here, one may remember Einstein's famous saying [2],

"As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality." 

• If we assume x, t as non-continuous entities (space, time as discrete), then:
  a) Assume x as continuous (as is done in all physics, excepting maybe string theory). To be sure, we use space/time as continuous concepts allowing use of tools such as modeling aspects of physical (apparently continuous) systems using differential equations to show their dynamic behaviors.
  Such equations yield continuous solution in x and t or in f(x,t), f supposed to be representing some physical quantity.
  b) Now, if reality is discrete, then clearly the differential, equations, allow as continuous distributions in x and t which reality does not allow, for a given energy range, for a dynamical system i.e., an non quantized reality is being mis-represented from a continuous model.
• Even if we need not know the lack or presence of the underlying reality, whether it be continuous or otherwise, the notion that the equations of the model will tell us what the case is is not correct. This is for obvious reasons that a model is as good as the assumptions it holds as true.
• A two fold error occurs if we allow the results (the solution) of the model because of its (false) assumptions, as above, and also because of another factor, viz, lack of incorporation of oneness or inter-relatedness.
  Keeping even a crude notion of the butterfly effect of chaos theory, the question arises as to the correction of not including the entire environment "around" a system. It may be asked as to whether essentially incorrect parts of a universe can be put together to represent the physical universe.

The lower spatial limit of the Universe maybe the Plank length = Ss (String size). Using this to combine with Calculus, in the form:

These ideas seem related to String Theory as a unifying theory of the Weak, Electromagnetic, Strong and Gravitational forces, eliminating the sharpness of Quantum foam and allowing for a smooth transition from Plank size (10 ^-35 meters) to the classical size we are intuitively more familiar with, say a brick.

Also, could integers, the set of numbers: {-N..., -3, -2, -1, 0, 1, 2, 3,...N}, where N is large and finite, be used alone by mathematics with no fractions? That is, to do away with Calculus. This need thinking over, surely, at the least! The Road to Reality by Roger Penrose, makes mention [3] of various ideas that have been proposed by thinkers, for e.g., by Ahmavaara (1965), who suggested that the real number system, fundamental to the mathematics of conventional physics, should be replaced by some field F_p, where p is some extremely large prime number..."Perhaps we should be seeking something of a character fundamentally different from the real-manifold setting of continuous spacetime of Einstein's theory and standard quantum mechanics depends upon."

Spatio temporal reality be defined by the Diritchlet look alike function, where D(x) = 1 if x = Natural numbers, and D(x) = 0 if x is irrational or rational with divisor not = 1.
Such a function may or may not satisfy the Fourier series requirements of a piecewise continuous function. If it does then such a function can be represented by Fourier series. This may serve as the link between the real ( discrete) world of physicists and the continuous (Platonic) world, of mathematicians such as Cantor. possible explaining why/how the equations of mathematics approximately models reality. Further it could provide a justification for continuing the use of continuous differential equations, suitable for modeling and manipulation, and the mapping to the discrete real world made finally for exact (precise) interpretation of the equations as they pertain to reality.
It may be possible to generate a purely algebraic system of mathematics, where "going over to the continuous world" is not required anymore.

Implications/Summary: If quantum theory does represent reality its discrete model should be used to represent gross matter too requiring a discrete mathematical tool. Further, once an appropriate math tool is used the Universe theories be put in place using inter-connectedness explicitly. This will require large funds and can be done privately by clever enterprise.

[1] The Road to Reality, Roger Penrose, Vintage Publications, 2005, Ch. 3, pg 62.
[2] Collected Quotes from Albert Einstein
[3] The Road to Reality, Roger Penrose, Vintage Publications, 2005, Ch. 33: More radical perspectives; twistor theory.

• God Created the Integers, Stephen Hawking, Chapter on G. F. B. Riemann: On the Hypothesis Which Lie at the basis of Geometry.
• Men of Mathematics, E. T. Bell, Chapter on Kronecker, The Doubter.
  Note: The present topic seems to lead to "related" Fourier Series notions. So reference to these series is made below:
  
• Erwin Kreyszig, Advanced Engineering Mathematics: Part C, Fourier series and Partial Differential Equations, 8th Edition.
• How does Maths work so well?
• What is the solution to Zeno's paradoxes?
• Truth and Knowledge © Nandakishore Shrikhande 2008