64
A. F. BARTHOLOMAY
information about nature and her basic mechanisms unless they have been
deduced mathematically from basic principles, which are, for governing the
behaviour of atoms and moleculres, the statistical or stochastic laws of quantum
theory. In biology there is neither place nor need for extensions of BOHR'S
complementarity philosophy, originally proposed to overbridge the gap between the two physical pictures of matter, viz. wave and particle, because in biology no such gap between different pictures ofliving matter exists. I am therefore
convinced that in all branches of science, including biology and psychology,
no mathematical model of any phenomenon can be considered as being a
theory of that phenomenon in the scientific sense of the word unless the model
is based upon and directly derived from quantum theory. Curves expressing
macroscopic phenomena can only be considered as having heuristic value.
BARTHOLOMAY (final comment given by correspondence):
To WALTER: My idea in delineating two main components of random error
was not to suggest either that they are independent (or separable) or the
opposite, but that in dealing with experimental data we should consider the
possibility that in certain cases unusual or unexpected random fluctuations
may not signify necessarily a poorly executed experimental run; the explanation
may be deeper than this.-To ARLEY (first comment): The fact that the standard validation of any method of chemical determination· rests on experiments
designed to estimate statistically the repeatability, reproducibility and precision
indicates that the observer is in a position to exert some measure of control
over random fluctuations. Certainly human error, coupled with instrumental
imprecision, are important factors contributing to the random features, apart
from inherent random aspects of the chemical mechanism itself. It is in this
sense that I refer to the former as "controllable". I do not mean to imply that I
consider that they are in any case controllable to the extent of being eradicable
in any practicable sense. If this were so, then one could examine the residual
inherent components in the ordinary experimental in vitro sense. The fact that
they appear not to be is the reason that I have introduced the in nllmero approach,
using the computer to simulate various patterns of random fluctuation as a
means of perhaps gaining a little additional insight into the whole complicated
question.-To BERGNER and ARLEY: In my paper referred to by BERGNER
(Bull. Math. Biophys. 21, (363 1959)) I derived a very simple statisticalformula
for estimating the stochastic parameter fl which corresponds to the classical
first-order rate constant k. In applying such a formula directly to data, one is
guilty of ignoring the presence of extraneous or "controllable" random fluctuations. On the other hand the usual regression methods for estimating k
directly from kinetic data ignore the presence of inherent fluctuations. In the
various cases treated in that paper it was of some interest to note that the
estimates of k and fl by these different rationales turned out to be indistinguishable from a statistical point of view. I therefore drew an heuristic inference
from such observations that the formula in the paper might be utilized as a
convenient, simple, numerical way (indirectly) of avoiding the more cumbersome classical curve-fitting and regression methods of estimating k.-To
KRUGER: I am afraid that I disagree with his comment that the starting point
for biomathematics is a mathematical description inevitably of macroscopic
phenomena; the body of my text should make this abundantly clear.-To
ARLEY: With all of its progress over the past two decades, biology has not
yet advanced to that same point as physics where a transition from classical
deterministic to statistical physics (and stochastic approaches) was seen as
A. F. BARTHOLOMAY
information about nature and her basic mechanisms unless they have been
deduced mathematically from basic principles, which are, for governing the
behaviour of atoms and moleculres, the statistical or stochastic laws of quantum
theory. In biology there is neither place nor need for extensions of BOHR'S
complementarity philosophy, originally proposed to overbridge the gap between the two physical pictures of matter, viz. wave and particle, because in biology no such gap between different pictures ofliving matter exists. I am therefore
convinced that in all branches of science, including biology and psychology,
no mathematical model of any phenomenon can be considered as being a
theory of that phenomenon in the scientific sense of the word unless the model
is based upon and directly derived from quantum theory. Curves expressing
macroscopic phenomena can only be considered as having heuristic value.
BARTHOLOMAY (final comment given by correspondence):
To WALTER: My idea in delineating two main components of random error
was not to suggest either that they are independent (or separable) or the
opposite, but that in dealing with experimental data we should consider the
possibility that in certain cases unusual or unexpected random fluctuations
may not signify necessarily a poorly executed experimental run; the explanation
may be deeper than this.-To ARLEY (first comment): The fact that the standard validation of any method of chemical determination· rests on experiments
designed to estimate statistically the repeatability, reproducibility and precision
indicates that the observer is in a position to exert some measure of control
over random fluctuations. Certainly human error, coupled with instrumental
imprecision, are important factors contributing to the random features, apart
from inherent random aspects of the chemical mechanism itself. It is in this
sense that I refer to the former as "controllable". I do not mean to imply that I
consider that they are in any case controllable to the extent of being eradicable
in any practicable sense. If this were so, then one could examine the residual
inherent components in the ordinary experimental in vitro sense. The fact that
they appear not to be is the reason that I have introduced the in nllmero approach,
using the computer to simulate various patterns of random fluctuation as a
means of perhaps gaining a little additional insight into the whole complicated
question.-To BERGNER and ARLEY: In my paper referred to by BERGNER
(Bull. Math. Biophys. 21, (363 1959)) I derived a very simple statisticalformula
for estimating the stochastic parameter fl which corresponds to the classical
first-order rate constant k. In applying such a formula directly to data, one is
guilty of ignoring the presence of extraneous or "controllable" random fluctuations. On the other hand the usual regression methods for estimating k
directly from kinetic data ignore the presence of inherent fluctuations. In the
various cases treated in that paper it was of some interest to note that the
estimates of k and fl by these different rationales turned out to be indistinguishable from a statistical point of view. I therefore drew an heuristic inference
from such observations that the formula in the paper might be utilized as a
convenient, simple, numerical way (indirectly) of avoiding the more cumbersome classical curve-fitting and regression methods of estimating k.-To
KRUGER: I am afraid that I disagree with his comment that the starting point
for biomathematics is a mathematical description inevitably of macroscopic
phenomena; the body of my text should make this abundantly clear.-To
ARLEY: With all of its progress over the past two decades, biology has not
yet advanced to that same point as physics where a transition from classical
deterministic to statistical physics (and stochastic approaches) was seen as