I am a Master’s degreed industrial engineer. I make my living by using a problem-solving method called Six Sigma. In short, this is a detailed problem solving methodology based on statistical methods.
I spend my days looking at manufacturing and service processes and analyzing their performance with 2-sample t-tests, ANOVA tables, Chi-squared tables, Factoral Experiments, and many other scientific methods of designed experimentation. I have grown to see many things in life through the window of hypothesis testing, the most often used scientific research method. I have a supreme respect for the scientific method, but there are things that science cannot prove. It must assume certain foundational principles.
Science assumes an external world that is orderly. By orderly, I mean a world that is organized according to basis principles. This assumption cannot be proven by science because the only way that science can explain order is to find more order. The law or theory is explained by another law or theory, but science has no explanation for the order we find other than to promote it as a brute fact. Using order to prove order is a viciously circular form of argument.
Science assumes that we can have knowledge of the world, that is, we can know truth that corresponds to reality. This is an abstract concept that a concrete field like science cannot explain. How do we show this to be true by examining the things we find and applying deductive reasoning? We cannot prove that we can have knowledge of the world by appealing to knowledge of the world. That would also be viciously circular. We have to turn to philosophy for an explanation (see here).
Science fares no better when it comes to other abstract principles like the laws of logic or mathematics, or even morality. You simply cannot prove an abstract, nonmaterial principle by observing the material world. You have to assume that these principles are true before you can use them to analyze experiments.
Science must also assume the closely related issue of the basic reliability of our senses. We must trust that our senses of smell, touch, sight, and hearing give us useful information about the world. We must be able to gain true knowledge of the world by our senses to try to learn specific truths from general inquiry.
Science cannot affirm this idea because an evolutionary process of development would not necessarily lead to true knowledge of the world around us. Evolution is a process that yields senses that allow for us to survive certain situations, but we could have untrue beliefs that guaranteed our survival in some situations. It doesn’t matter whether we know a tiger can eat us or that we think tigers look better from a distance. As long as we run, we survive. (See Alvin Plantinga’s Warrant and Proper Function here or a link to some of the relevant material here.)
Science is not a method that can ‘stand on its own two feet.’ We must assume many foundational principles before science can give us any meaningful information at all whatsoever.
[Source for much of this argument: J. P. Moreland, “Christianity and The Nature of Science,” lectures for the Biola University Defending Your Faith Series, see here.]