If a property in $\mathbb{N}$ is true up to $10^{47}$ are there reasons to think it is probably true in all $\mathbb{N}$? You have probably heard at some point statements like that the twin prime conjecture (namely that $2$ is an infinitely ocurring prime gap) is "probably" or "almost certainly" true. Same goes for a number of other problems in number theory asserting properties that we believe are satisfied by all natural numbers. An argument that seems to give ground to such beliefs is the observation that said conjecture holds up to a certain large number. 


  My question would be if there some solid argument to think that a given conjecture is "probably" true, given that we know it is true up to some large number (as big as you wish but fixed). 


The answer upfront seems to be no, and still the thought is so unavoidable I wonder if someone has thought of an argument to justify it. 

EDIT: Do not take the question rigurously. I was looking for a broad take on this. Think for example of the case of the twin prime conjecture. Of course I know there are properties satisfied by numbers only above a given number (for example the property of being bigger than said number). Be imaginative. Tell me results or arguments that you think might relate. Tell me about conjectures where, while not knowing if they hold, this line of thought is somehow justified and why. 

EDIT $2$: Case where this question would apply. Let's say we conjecture some property of the natural numbers. About the size of a possible minimum counterexample we are not able to find anything besides that it must be greater than $10^{47}$. At this point the thought might reach our minds that the conjecture is "probably" true. Now, is there any convincing argument that justifies this thought?
No.

In fact, most positive integers are larger than $10^{47}$. An example of a property that is true for integers below $10^{47}$ but not true for integers larger than $10^{47}$ is being smaller than $10^{47}$.

A more apt but common example can be found in number theory. The number of primes up to $x$ is denoted by $\pi(x)$, and is commonly approximated by the logarithmic integral 
$$\text{Li}(x) = \int_2^x \frac{1}{\ln t}dt$$ Asymptotically, they are equal. It was believed for a long time that $\pi(n) &lt; \text{Li}(n)$ for all $n$ (both Gauss and Riemann conjectured this). But it is not true. The first counterexample is thought to be somewhere around $10^{300}$.
$10^{10^{47}}$ have all digits at zero starting from unity up to the $10^{47th}$. Still, I advice you to verify one more step ;-)