Embarking on a water-bound adventure can be the highlight of any vacation. Whether it’s meandering through the picturesque canals of Amsterdam, marveling at the teeming wildlife of the Galapagos, or soaking in the grandeur of Sydney Harbour, boat tours offer an unparalleled perspective of some of the world’s most stunning locations.

However, selecting the perfect boat tour company to facilitate this adventure can be a daunting task. This piece aims to guide you through this complex process in a logical, step-by-step fashion, integrating principles from various fields including history, law, economics, mathematics, and social sciences.

The first factor to consider is the historical reputation of the company. A long-standing boat tour company with an established track record is more likely to provide a reliable and enjoyable experience. The Lindblad Expeditions, for example, has been operating since 1979 and has garnered a reputation for their unique expeditions to some of the world's most remote waters. This factor, however, should not discourage you from considering newer companies which may offer innovative and refreshing approaches to boat touring.

The legal aspect is another critical consideration. All reputable boat tour companies should adhere to maritime law and safety regulations. Always verify if the company has the necessary licenses and insurance. Companies without these can expose you to significant risks and liabilities. Also, inquire about their safety protocols and emergency preparedness measures, which plays a vital role in the event of unexpected incidents.

Economic considerations also come into play when hiring a boat tour company. Prices for boat tours can vary significantly, depending on the length of the tour, the type of vessel, and the destinations covered. To make an informed decision, utilize the principle of opportunity cost. This economic concept suggests that for each potential tour, you should consider what you are potentially giving up (in terms of money and time) versus what you are gaining. Use this as a comparative tool to guide your decision.

One cannot underestimate the role of mathematics and statistics when selecting a boat tour company. Look into data such as customer satisfaction rates, the number of successful tours conducted, or the ratio of tour guides to tourists. For instance, smaller tour groups often lead to more personalized experiences, so consider companies that maintain lower tourist-to-guide ratios.

The choice of the boat tour company should also reflect social dynamics. This encompasses the cultural sensitivity of the company, the inclusivity, and the environmental impact of their operations. Companies that respect local cultures, promote inclusivity, and strive for eco-friendly practices are more likely to provide enriching and guilt-free experiences.

Technological innovations have also entered the realm of boat tours, revolutionizing the way they operate. Some companies now offer virtual reality tours, while others have developed apps to enhance the learning experience. While this can certainly add a new dimension to the tour, the underlying tradeoff is between the authenticity of the experience and the technological enhancement. One must ponder, does a VR headset showing the underwater marine life detract from the charm of peering over the boat's edge into the water?

Finally, the choice of the boat tour company is also a reflection of one's personal preferences. Some people may prefer an intimate sunset cruise, while others may seek an adrenaline-pumping speedboat ride. Therefore, despite all the aforementioned analytical considerations, the final decision is inherently subjective and personal.

While this guide provides an intellectually rigorous approach to selecting a boat tour company, it is essential to remember that the ultimate goal is enjoyment. So while you dissect the history, law, economics, mathematics, statistics, and social science aspects of your decision, don't lose sight of what truly matters - the anticipation of the spray of the sea, the wind in your hair, and the promise of unforgettable moments on the horizon.